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AN 


ELEMENTARY  TREATISE 


ARITHMETIC. 


PiaXCIPALLY  FROM  THE  ARITHMETIC 


S.  F.  LACROIX, 

M 

iiNO 


TRANSLATED    IVTO    ENGLISH    WITH    SUCH    ALTEKATIO.VS    AND 

AnOlTIONS  AS  WERE   FOUND  NECESSARY  IN  ORDER  TO 

ADAPT    IT    TO    THE    USE    OF    THE 

AMERICAN  STUDENT, 


CAMIJ RIDGE,  -N.  E. 


raruTED  hi  hilliari)  and  jiktcat-f,  at  thk  uyivEiisiTT  prms. 

Sold  by  W.  Hilliard,  Cambridge,  and  by  Curaniinjs  &  Hiiiiard, 
No.  1  Coriihill,  Boston. 

1818. 


DISTRICT  CF  MASSACHUSETTS,  TO  WIT : 

District  Clerk's  Office^ 
BE  IT  REMEMBERED,  That  on  the  twenty  fourth  day  of  August,  A.  D.  1818,  and  in  thfe 
forty  third  year  of  the   Independence  of  the  United  States  of  America,  Cummings  &  Hilliaid,  of 
th(  said  District,  have  deposited  in  this  office  the  title  of  a  Book,  the  right  whereof  they  claim  as 
proprietors,  in  the  words  folloving,  viz. 

"An  elementary  treatise  ou  Arithmetic,  taken  principally  from  the  arithmetic  of  S.  F.  Lacrois^ 
and  translated  into  English  with  such  alterations  and  additions  as  were  found  necessary  in  order  to 
adapt  it  to  the  use  of  the  American  student. 

In  conformity  to  the  Act  of  the  Congress  of  the  United  States,  entitled,  "An  Actfor  the  en- 
couragement of  learning,  by  securing  the  copies  of  Maps,  Charts,  and  Books,  to  the  Authors  and 
Proprietors  of  such  copies,  during  the  times  therein  mentioned  ;"  and  also  to  an  Act,  entitled,  «  An 
Act  supplementary  to  an  Act,  entitled.  An  act  for  the  encouragement  of  learning,  by  securing  the 
copies  ot  Maps,  Charts,  and  Books,  to  the  Autfiors  and  Proprietors  of  such  copies  daring  the  times 
therein  mentioned ;  and  extending  the  benefits  thereof  to  the  Arts  of  Designing,  Engraving  and 
Etching  Historical  and  other  Prints." 

.JNO.  W.  DAVIS, 
Clerk  of  the  District  of  Massnchusetts. 


ADVERTISEMENT. 

The  first  principles,  as  well  as  the  more  difficult  parts  of 
Mathematics,  have,  it  is  thought,  been  more  fully  and  clearly  ex- 
plained by  the  French  elementary  writers,  than  by  the  English ; 
and  among  these,  Ladroix  has  held  a  very  distinguished  place. 
His  treatises  have  been  considered  as  the  most  complete,  and  the 
best  suited  to  those  who  are  destined  for  a  public  education.  They 
have  received  the  sanction  of  the  Government,  and  have  been  adopt- 
ed in  the  principal  schools,  of  France.  The  following  translation  is 
from  the  thirteenth  Paris  edition.  The  original  being  written  with 
reference  to  the  new  system  of  weights  and  measures,  in  which 
the  different  denominations  proceed  in  a  decimal  ratio,  it  was 
found  necessary  to  make  considerable  alterations  and  additions,  to 
adapt  it  to  the  measures  in  use  in  the  iTnitcd  States.  The  several 
articles  relating  to  the  reduction,  addition,  subtraction,  multiplica- 
tion, and  division,  of  compound  numbers,  have  been  written  anew  5 
a  change  has  been  made  in  many  of  the  examples  and  questions, 
and  new  ones  have  been  introduced  after  most  of  the  rules,  as  an 

exercise  for  the  learner. 

JOHN  FARRAR, 

Professor  of  Mathematics  and  Natural  Phijjjs* 
ophy  in  the  University  at  Cambridge. 

Cambridge,  Mg.  1818. 


Ml914y9 


CONTENTS. 

sneral  remarks  on  the  different  kinds  of  magnitude  or  quantity      1 
Of  number  ---...  ibid. 

Of  spoken  numeration  -  ,  -  -  .  g 

banner  of  representing  numbers  by  figuresi  or  written  numera- 
tion -.-...-4 
Of  reading  numbers  -  -  .  -  .  q 
Of  abstract  and  concrete  numbers            -            -            -            -    7 

Of  Addition. 

Of  the  principles  on  which  addition  is  founded  -  -         7 

General  rule  for  performing  addition  -  -  -  9 

Of  Subtraction, 

Of  the  principles  on  which  it  is  founded  -  -  -        9 

Explanation  of  the  terms,  remainder,  excess,  and  difference  -      10 
General  rule  for  subtraction             -            -            -             -  12 

Method  of  proof  for  addition  and  subtraction  -  .  IS 

Of  Multiplication. 

The  origin  of  multiplication  ....  14 

An  explanation  of  the  terms,  multiplicand,  multiplier,  product, 
a.nd  factors  -  -  -  -  -         -        15 

Of  the  principles  on  which  multiplication  is  performed         -        ibid. 
The  table  of  Pythagoras,  containing  the  products  of  any  two  fig- 
ures -  -  -  -  -  >  -16 

Formation  of  this  table  -  -  -  -  -      ibid. 

Remarks,  from  which  it  is  inferred,  that,  a  change  in  the  order  of 
the  factors  does  not  affect  the  product  -  -  17 


vi  Contentiv 

Rule  for  multiplying  a  number,  consisting  of  several  figures,  by  a 
single  figure  -  -  -  -  -  -18 

Tomultiply  bylO,  100,  1000,&c.  -  -  -        -         19 

Rule  for  multiplying  by  a  number  consisting  of  a  single  digit  and 
any  number  of  ciphers  -  -  -         -  -        20 

General  rule  for  multiplication  -  -  -  -        21 

Manner  of  abridging  the  process,  when  both  factors  are  termin- 
ated by  ciphers  -  -  -  _         _        .        gg 

Of  Division. 

The  origin  of  division            -            -            -            -  •         -  22 

Explanation  of  the  terms,  dividend,  divisor,  and  quotient         -  23 

Of  the  principles  on  which  division  is  founded  -  -  ibid. 
JVf  ode  of  proceeding,  when  the  divisor  consists  of  several  figures     27 

,  General  rule  for  division              -             -             -             -  28 

Method  of  abbreviating  the  process  of  division            -            -  29 

When  both  the  divisor  and  dividend  have  ciphers  on  the  right  30 

Multiplication  and  division  mutually  prove  each  other         -  ibid. 

Of  Fractions. 

The  origin  of  fractions             -            -            -            -            -  SO 

The  manner  of  reading  andf  writing  fractions  -  -  32 
An  explanation  of  the  terms  numerator  and  denominator  -  ibid. 
Of  the  changes  which  a  fraction  undergoes,  by  the  increase  or 

diminution  of  one  of  its  terms  -  -  -  -  33 
A  Table  representing  the  changes  which  take  place  in  a  fraction, 

by  the  multiplication  or  division  of  either  of  its  terms  -  34 
The  value  of  a  fraction  not  altered  by  multiplying  or  dividing 

both  its  terms  by  the  same  number  *  -         -         ibid. 

To  simplify  a  fraction  without  altering  its  value        -        -       -  35 

The  greatest  common  divisor  of  two  numbers             -             -  36 

General  rule  for  finding  the  greatest  common  divisor         -         -  38 

To  distinguish  the  numbers  divisible  by  2,  5  or  3          -          -  39 

Of  prime  numbers              -             -              .             -            -  40 

General  signification  of  the  term  multiplier              -            -  41 

To  multiply  a  whole  number  by  a  fraction             -             -  42 

To  find  the  whole  number  contained  in  a  fraction         -         -  43 

To  reduce  a  whole  number  to  a  fraction            -            -         -  44 


Contents,  vii 


To  multiply  one  fraction  by  another  -  .  -  ibid. 
Of  compound  fractions  -  -  -  -  45 
Of  division  in  general  -  -"  /  ■  -  "  ^^^^' 
Of  the  division  of  a  whole  number  by  a  fraction  -  -  46 
To  divide  one  fraction  by  another  .  .  -  ibid. 
Of  the  addition  and  subtraction  of  fractions  -  -  47 
To  reduce  fractions  to  a  common  denominator  -  -  ibid. 
Of  the  addition  and  subtraction  of  mixed  numbers  -  -  49 
The  product  of  several  factors  not  changed,  by  changing  the  or- 
der in  which  they  are  multiplied            -            -            -  50 

Of  Decimal  Fractions. 

The  origin  of  decimal  fractions            -            -            -            -  51 

The  manner  of  reading  and  writing  decimals                          -  52 

A  number  containing  decimals,  not  altered  by  annexing  ciphers  54 

Addition  of  decimals             .             .             .             .           .  Jbij, 

Subtraction  of  decimals  -  -  -  -  '53 

The  effect  of  changing  the  place  of  the  decimal  point        -        -  56 

To  multiply  a  number  containing  decimals,  by  a  whole  number  57 

The  multiplication  of  one  decimal  by  another         -         -         -  58 

To  divide  a  decimal  number  by  a  whole  number          -          -  59 

To  divide  one  decimal  by  another         -----  ibid. 

Method  of  approximating  the  quotient  of  a  division  by  decimals  ibid. 
JVote. — Method  of  finding  the  value  of  the  quotient  of  a  division 

.in  fractions  of  a  given  denomination            -            -          .  60 
To  reduce  vulgar  fractions  to  decimals            -            -            -  61 
JVote. — On  the  changing  of  one  fraction  to  another  of  a  lower  de- 
nomination            -            -            -            -            -          -  62 

Of  periodical  decimals            -            -          -            -            -  6S 


Tables  of  Coin,  Weight  and  Measure, 


Federal  Money             -  -            w            -            *            65 

English  Money             -  -            -            .            -            66 

Troy  Weight           ...  ...          ibid. 

Apothecaries  Weight  -            -            -          -        .        ibid. 

Avoidupois  Weight  -            -            -             .          .          67 

Dry  Measure             -  -             .            .            .            ibid. 

Ale  and  Beer  Measure  ....           j^d.' 


till  Contents* 

Wine  Measure  -             -           -            -  -  68 

Cloth  Measure  -            ...  -  Jbid. 

Long  Measure  -  -            -            .          -  _  ibid. 

Time             -  -            -             -            .  -  ibid. 

Reduction. 

To  reduce  pounds  and  shillings  to  pence  &c.  -  -         69 

To  reduce  shillings  to  pounds  -  -  -  -  70 

To  reduce  other  denominations  of  money,  weight,  and  measure      71 
To  reduce  a  compound  number  to  the  lowest  denomination  con- 
tained in  it  -  -  -  -  -  -  72 

To  reduce  a  number  from  a  lower  denomination  to  a  higher    -    ibid. 
To  reduce  the  several  parts  of  a  compound  number  to  a  fraction 

of  the  highest  denomination  contained  in  it  -  -  73 

To  find  the  value  of  a  fraction  of  a  higher  denomination,  in  the 

terms  of  a  lower  .  .  .  -  _         ibid. 

To  reduce  the  several  parts  of  a  compound  number  to  a  decimal 

of  the  highest  denomination  contained  in  it  -  -         75 

To  reduce  a  decimal  of  a  higher  denomination  to  a  lower        -        76 
To  convert  shillings,  pence  and  farthings,  to  the  decimal  of  a 
pound  -  -  -  -  -  -  7f 

To  convert  the  decimal  of  a  pound  to  shillings  &c.         -        -     ibid. 
To  reduce  numbers  from  one  denomination  to  another,  when  the 
two  numbers  are  not  commensurable  -  -        -         78 

Of  Compound  JWmbers. 

General  rule  for  the  addition  of  compound  numbers         -  -       80 

Method  of  proving  the  addition  of  compound  numbers  -          81 

Of  the  subtraction  of  compound  numbers            -            -  -        82 

To  prove  subtraction  of  compound  numbers            -  -            83 

Multiplication  of  compound  numbers            -            -  -           84 

General  rule  for  the  multiplication  of  compound  numbers  -       85 

Mode  of  proceeding,  when  the  multiplier  exceeds  12      -  -      ibid. 
Of  duodecimals            ------         86 

General  rule  for  the  multiplication  of  duodecimals         -  -        87 

Of  the  division  of  compound  numbers            -            -  -         89 

General  rule  for  compound  division            .            -  -            90 
Method  of  operation,  when  the  divisor  is  large,  and  resolvable 

into  two  or  more  factors            -            -            -  -         ibid. 


Contents,  ix 

Process,  when  the  divisor  cannot  be  8o  resolved          -          -  91 

Multiplication  and  division  mutually  prove  each  other        -  ibid. 

Of  Proportion, 

A  development  of  the  principles  on  which  the  rules  of  proportion 

are  founded            -            -            -            -            -            -  92 

Of  the  nature  of  ratios            -            -            ...  93 

Explanation  of  the  terms,  relation  and  ratio         -         -         -  ibid. 

Of  the  term  proportion            -            -            -            -            -  94 

Of  the  terms  antecedent  and  consequent            -            -          -  ibid. 
Of  the  equality  of  the  product  of  the  means  to  that  of  the,  extremes    95 

Transposition  of  the  terms  of  a  proportion             -            -  ibid. 

To  obtain  any  one  term  of  the  proportion  from  the  other  three  96 

The  Rule  of  Three             -             -            -            .            -  97 

Rules  for  the  stating  of  questions             ...  ibid. 

An  elucidation  of  these  rules            -            -            -            -  98 

General  rule  for  solving  all  questions  in  proportion        -        -  99 

Examples  for  illustration            -            ....  ibid. 

Questions  for  practice            -            -           -            -          .  lOO 

Compound  Proportion. 

The  Double  Rule  of  Three            -            -            -            -  103 

Of  the  principles  on  which  it  is  founded            ...  ibid. 

These  principles  illustrated  by  examples            ...  104 

Of  the  compounding  of  ratios            -            -            -            -  106 

General  rule  for  solving  questions  in  compound  proportion      -  107 

Examples  tor  practice            -            -            .            .        .  ibid. 

Fellowship, 

The  use  of  the  rule,  and  the  principles  en  which  it  is  founded  180 
Commercial  use  of  the  terms,  capital  or  stock  and  dividend  -  109 
Examples  illustrative  of  the  principles  of  fellowship  -  -  ibid. 
Of  e^Mic?(^«r«wce  in  numbers,  or  arithmetical  ratios  -  -  110 
J^ote. — Distinction  between  geometrical  and  arithmetical  pro- 
portion            ibid. 

Manner  of  writing  numbers  in  e^ruiiijference         -        -        -  111 

Questions  for  practice           .....  ibid. 
b 


t  Contents, 

Of  Alligation. 

The  principles  of  medial  alligation  explained            -  113 

Illustrated  by  examples            .            _            -            .        .  ibid. 

Alligation  alternate  explained         -           -            -           -  114 

Examples  for  illustration             -            -             -            -  115 

Miscellaneous  €(uestions  117 
Appendix, 

Tables  of  various  weights  and  measures            -          -         -  119 

New  French  weights  and  measures            .            ,            -  ibid. 

Reasons  for  adopting  the  decimal  gradation            -            -  ibid. 

The  measures  of  length             -            -            ...  jbid. 

The  measures  of  capacity             -             -             -            -  120 

Weights            -             -             ...            -  ibid. 

Land  Measure              -             _             _             -             -  121 

The  division  of  the  circle             .            .            -            ■  ibid. 

The  decimal  system  of  coin            _            -            .            -  ibid. 

Divisions  of  time              .             .            .            -            -  ibid. 

Scripture  long  measure              -              -             -             *  122 

Grecian  long  measure  reduced  to  English             -            -  ibid. 

Jewish  long  or  itinerary  measure              ...  12S 

Roman  long  measure  reduced  to  English             -             -  ibid. 

Attic  dry  measure  reduced  to  English           ,           .          -  124 
Attic  measures  of  capacity  for  liquids  reduced  to  English  wine 

measure            ,._-.--  ibid. 
Measures  of  capacity  for  liquids  reduced  to  English  wine  mea- 
sure           -            -            -            -            -        -         .  ibid. 

Jewish  dry  measure  reduced  to  English           -            -          -  125 
Jewish  measure  of  capacity  for  liquids  reduced  to  English  wine 

measure            _>_.--  ibid. 

Roman  dry  measure  reduced  to  English             -              -  ibid. 
Of  the  principal  gold  and  silver  coins,  containing  their  weights, 
fineness,  pure  contents,  current  value,  &c.        .        -        -      126 


Eccplanation  of  the  Roman  JSTumeraU. 

One  I 

Two  II* 

Three  HI 

Four  IVf 

Five  V 

Six  Vlt 

Seven  VII 

Eight  VIII 

Nine  IX 

Ten  X 

Twenty  XX 

Thirty  XXX 

Forty  XL 

Fifty  L 

Sixty  LX 

Seventy  LXX 

Eighty  LXXX 

Ninety  XC 

Hundred  C 

Two  hundred  CC 

Three  hundred  CCC 

Four  hundred  CCCC 

•  As  often  as  any  character  is  repeated,  so  maiiy  times  its  value  is  re- 
peated. 

t  A.  less  character  before  a  greater  diminishes  its  value. 
?  A  less  character  after  a  greater  increases  its  value* 


xii 


Bommi  JVhimerals, 


Five  hundred 
Six  hundred 
Seven  hundred 
Eight  hundred 
Nine  hundred 
Thousand 
Eleven  hundred 
Twelve  hundred 
Thirteen  hundred 
Fourteen  hundred 
Fifteen  hundred 
Two  thousand 
Five  thousand 
Six  thousand 
Ten  thousand 
Fifty  thousand 
Sixty  thousand 
Hundred  thousand 
Million 
Two  millions 


D  or  10* 

DC 

DCC 

DCCC 

DCCCC 

M  or  ClOt 

MC 

MCC 

MCCC 

MCCCC 

MD 

MM 

IOO:orVt 

VI 

XorCCIOO 

lOJJ 

LX 

C  or  CCCIOOO 

M  or  CCCCIOOOJ 

MM 


*  For  every  o'affixed  this  becomes  ten  times  as  many 

f  For  every  C  and  0  put  one  at  each  end,  it  is  increased  ten  tiroes. 

i  A  line  over  anjr^ number  increases  it  1000  fold. 


ELEMENTARY  TREATISE 


ARITHMETIC. 


J^meration, 

1.  A  COMPARISON  of  the  different  objects,  that  come  within  the 
reach  of  our  senses,  soon  leads  us  to  perceive,  that,  in  all  these 
objects,  there  is  an  attribute,  or  quality,  by  which  they  can  be 
supposed  susceptible  of  increase  or  diminution  ;  this  attribute  is 
magnitude.  It  generally  appears  in  two  different  forms.  Some- 
times as  a  collection  of  several  similar  things,  or  separate  parts, 
and  is  then  designated  by  the  word  number. 

Sometimes  it  presents  itself  as  a  whole,  without  distinction  of 
parts ;  it  is  thus,  that  we  consider  the  distance  between  two 
points,  or  the  length  of  a  line  extending  from  one  to  the  other, 
as  also  the  outlines  and  surfaces  of  bodies,  which  determine 
their  figure  and  extent,  and  finally  this  extent  itself. 

The  proper  characteristic  of  this  last  kind  of  magnitude,  is 
the  connexion  or  union  of  the  parts,  or  their  continuity ;  whilst 
in  number  we  consider  how  many  parts  there  are ;  a  circum- 
stance to  which  the  word  quantity  at  first  had  relation,  though 
afterwards  it  was  applied  to  magnitude  in  general,  magnitude  con- 
sidered as  a  whole  being  called  continued  quantity ,  to  distinguish 
it  from  number,  which  is  called  discrete,  or  discontinued,  quantity^ 

2.  All  that  relates  to  magnitude  is  the  object  of  mathematics  ; 
numbers,  in  particular,  are  the  object  of  arithmetic. 

Continued  magnitude  belongs  to  geometry,  which  treats  of  the 
properties  presented  by  the  forms  of  bodies,  considered  with 
regard  to  their  extent. 

3.  Number,  being  a  collection  of  many  similar  things,  or  many 

1 


2  Arithmetic. 

distinct  parts,  supposes  the  existence  of  one  of  these  things,  or 
parts,  taken  as  a  term  of  comparison,  and  this  is  called  unity. 

Tlie  most  natural  mode  of  forming  numbers  is,  to  begin  with 
joining  one  unity  to  another,  then,  to  this  sum  another  j  and 
continuing  in  this  manner,  we  obtain  collections  of  units,  which 
are  expressed  by  particular  names  ;  the  whole  of  these  names, 
which  varies  in  different  languages,  composes  the  spoken  numera- 
tion. 

4.  As  there  are  no  limits  to  the  extension  of  numbers,  since 
however  great  a  number  may  be,  it  is  always  possible  to  add  an 
unit  to  it,  we  may  easily  conceive  that  there  is  an  infinity  of 
different  numbers,  and,  consequently,  that  it  would  be  impossible 
to  express  them  in  any  language  whatever,  by  names,  that  should 
be  independent  of  each  other. 

Hence  have  arisen  nomenclatures,  in  which  the  object  has 
been,  by  the  combinations  of  a  small  number  of  words,  subject 
to  regular  forms,  and  therefore  easily  remembered,  to  give  a 
great  number  of  distinct  expressions. 

Those,  which  are  in  use  in  the  [English  language,]  with  few 
exceptions,  are  derived  from  the  names  assigned  to  the  nine  first 
numbers  and  those  afterwards  given  to  the  collections  of  ten, 
a  hundred,  and  a  thousand  imits. 

The  units  are  expressed  by 

one,  two,  three,  four.  Jive,  six,  seven,  eight,  nine. 

The  collections  of  ten  units,  or  tens,  by 

ten,  twenty,  thirty,  forty, fifty,  sixty,  seventy,  eighty,  ninety. 

The  collections  of  ten  tens,  or  hundreds,  are  expressed  by 
names  borrowed  from  the  units ;  thus  we  say, 

hundred,  two  hundred,  three  hundred,  .....  nine  hundred. 

The  collections  of  ten  hundreds,  or  thousands,  receive  their 
denominations  from  the  nine  first  numbers  and  from  the  collec- 
tions of  tens  and  hundreds  ;  thus  we  say 

thousand,  two  thousand  ....  nirie  thousand, 

ten  thousand,  twenty  thousand,  ^c. 

hundred  thmisand,  two  hundred  thousand,  ^c. 

The  collections  of  ten  hundred  thousands,  or  of  thousands 
of  thousands,  take  the  name  of  millions,  and  are  distinguished, 
like  the  collections  of  thousands. 


Mimeration,  ^ 

The  collections  of  ten  hundreds  of  millions,  or  of  thousands  of 
millions,  are  called  hillionSf  and  are  distinguished,  like  the  collec- 
tions of  millions.! 

t  The  idea  of  number  is  the  latest  and  must  difficult  to  form.  Be- 
fore the  mind  can  arrive  at  such  an  abstract  conception,  it  must  be 
familiar  with  that  process  of  classification,  by  which  we  successively 
remount  from  individuals  to  species,  from  species  to  genera,  and  from 
genera  to  orders.  The  savage  is  lost  in  his  attempts  at  numeration, 
and  significantly  expresses  his  inability  to  proceed  by  holding  up  his 
expanded  fingers,  or  pointing  to  the  hairs  of  his  head. 

Nature  has  furnished  the  great  and  universal  standard  for  compu- 
tation in  the  fingers  of  the  hand.  All  nations  have  accordingly 
reckoned  hy  Jives  ;  and  some  barbarous  tribes  have  scarcely  advanc- 
ed any  further.  After  the  fi;igers  of  one  hand  had  been  counted  once, 
it  was  a  second  and  perhaps  a  distant  step  to  proceed  to  those  of  the 
other.  The  primitive  words  expressing  numbers  did  not  probably 
exceed  five.  To  denote  six,  seven,  eight  and  nine,  the  North  Amer- 
ican Indians  repeat  the  five  with  the  successive  addition  of  one,  two, 
three,  and  four ;  could  we  safely  trace  the  descent  and  affinity  of  the 
abbreviated  terms  denoting  the  numbers  from  five  to  ten,  it  seems 
highly  probable,  that  we  should  discover  a  similar  process  to  have 
taken  place  in  the  formation  of  the  most  refined  languages. 

The  ten  digits  of  both  hands  being  reckoned  up,  it  then  became 
necessary  to  repeat  the  operation.  Such  is  the  foundation  of  our  deci- 
mal scale  of  arithmetic.  Language  still  betrays  by  its  structure  the 
original  mode  of  preceding-  To  express  the  numbers  beyond  ten, 
the  Laplanders  combine  an  ordinal,  with  a  cardinal  digit.  Thus, 
eleven,  twelve,  &c.  they  denominate  second  ten  and  one,  second  ten 
and  two,  &c.  and  in  like  manner  they  call  twenty  one,  twenty  two, 
&c.  third  ten  and  one,  third  ten  and  two,  &c.  Our  term  eleven  is 
supposed  to  be  derived  from  ein  or  one,  and  lihen,  to  remain,  and 
to  signify  one,  leave  or  set  aside  ten.  Twelve  is  of  the  like  de- 
rivation and  means  two,  laying  aside  the  ten.  The  same  idea  is  sug- 
gested by  our  termination  ty  in  the  words  tiventy,  thirty,  &c.  This 
syllable  altogether  distinct  from  ten  is  derived  from  ziehen  to  draWf 
and  the  meaning  of  twenty  is,  strictly  speaking,  two  drawings,  that 
is,  the  hands  have  been  twice  closed  and  the  fingers  counted  over. 
After  ten  was  firmly  established,  as  the  standard  of  numeration,  it 


4  Jrithmetie. 

Each  of  the  names  just  mentioned  is  considered  as  forming  a 
unit  of  an  order  more  elevated  according  as  it  is  removed  from 
simple  unit.  The  names  ten  and  hundred  are  continually  re- 
peated and  we  have  no  occasion  for  new  names,  such  as  thou- 
sand, miUion,  UUim,  except  at  every  fourth  order.  The  same 
law  being  observed,  to  billions  succeed  trillions,  quadrUlionSf 
quintUlions,  &c.  each,  like  billions,  having  its  tens  and  hundreds. 

Numbers  expressed  in  this  manner,  when  more  than  one  word 
enters  into  the  enunciation  of  them,  are  separated  iato  their 
respective  ordei*s  of  units,  mentioned  above ;  for  instance,  the 
number  expressed  hy  Jive  hundred  thousand  three  hundred  and  two, 
is  separated  into  three  parts,  \iz,Jive  hundreds  of  thousands,  three 
hundreds  of  simple  units,  and  two  of  these  units. 

5,  The  length  of  the  expression,  written  in  words,  when  the 
numbers  were  large,  occasioned  the  invention  of  characters,  ex- 
clusively adapted  to  a  shorter  representation,  and  hence  origi- 
nated the  art  of  expressing  numbers  in  writing  by  these  charac- 
ters called /^ures,  or  written  numeration. 

The  laws  of  the  written  numeration,  now  used,  are  very  anal- 
ogous to  those  of  the  spoken  numeration.  In  it  the  nine  first 
numbers  are  each  represented  by  a  particular  character,  viz. 

1234567  89 

one,  two,  three,  four,  five,  six,  seven,  eight,  nine. 
"When  a  number  consists  of  tens  and  units,  the  characters  repre- 
senting the  number  of  each  are  written  in  order  from  left  to 
right,  beginning  with  the  tens.  The  number  forty-seven,  for 
instance,  is  written  47  j  the  first  figure  on  the  left,  4,  denotes  the 
four  tens,  and  consequently  a  value  ten  times  greater  than  it 
would  have  standing  alone  j  while  the  figure  7,  placed  on  the 

seemed  the  most  easy  and  consistent  to  proceed  by  the  same  repeated 
composition.  Both  hands  being  closed  ten  times  would  carry  the 
reckoning  up  to  a  hundred.  This  word,  originally  hund,  is  of  uncer- 
tain derivation ;  but  the  term  thousand  which  occurs  at  the  next  stage 
of  the  progress,  or  the  hundred  added  ten  times  is  clearly  traced  out, 
being  only  a  contraction  of  duis  hund,  or  twice  hundred,  that  is,  the 
repetition,  or  collection  of  hundreds.  See  Edinburgh  Review,  vol. 
xviu.  art,  viT. 


'■  J^iimeration.  5 

right,  exjiressing  seven  units,  possesses  only  its  original  value. 
In  the  number  thirty-three,  which  is  written  33,  we  see  the 
figure  3  repeated,  but  each  time  with  a  different  value;  the 
value  of  the  3  on  the  left  is  ten  times  greater  than  the  value  of 
that  on  the  right. 

This  is  the  fundamental  law  of  our  written  numeration,  that 
a  remavah  of  one  place,  towards  the  left  increases  the  value  of  a 
figure  ten  times. 

If  it  were  required  to  express  fifty,  or  five  tens,  as  there  are 
no  units  in  this  number,  there  would  be  nothing  to  write  but  the 
figure  5,  and  consequently  it  would  be  necessary  to  show,  by 
some  particular  mark,  that  in  the  expression  of  this  number,  the 
figure  ought  to  occupy  the  first  place  on  the  left.  To  do  this  we 
place  on  the  right  the  character  0,  cipher  or  nought ,  which  of 
itself  has  no  value,  and  serves  only  to  fill  the  place  of  the  units, 
which  are  wanting  in  the  enunciation  of  the  proposed  number. 
6.  Thus  with  ten  characters,  by  means  of  the  rule  before  laid 
down  concerning  the  value  which  figures  assume,  according  to 
the  places  they  occupy,  we  can  express  all  possible  numbers. 

"With  two  figures  only,  we  can  write  all,  as  far  as  to  nine  tens 
and  nine  units,  making  99,  or  ninety  nine.  After  this  comes  the 
hundred,  which  is  expressed  by  the  figure  1,  put  one  place  far- 
ther towards  the  left,  than  it  would  be,  if  used  to  express  tens 
only ;  and  to  denote  this  place,  two  ciphers  are  placed  on  the 
right,  making  100. 

The  units  and  tens,  afterwards  added  to  form  numbers  greater 
than  100,  take  their  proper  places  ;  thus  a  hundred  and  one  will 
be  written  in  figures  101 ;  a  hundred  and  eleven.  111.  Here  the 
same  figure  is  three  times  repeated,  and  with  a  different  value 
each  time ;  in  the  first  place  on  the  right  it  expresses  an  unit, 
in  the  second,  a  ten,  in  the  third,  a  hundred.  It  is  the  same 
with  the  number  222,  333,  444,  &c.  Thus,  in  consequence  of 
the  inile  laid  down  before  when  speaking  of  units  and  tens,  the 
same  figure  expresses  units  ten  times  greater,  in  proportion  as  it  is 
removed  from  right  to  left,  and  by  a  simple  change  of  place,  acquires 
the  power  of  representing  successively,  all  the  different  collections  of 
units,  tvhich  can  enter  into  the  expression  of  a  number. 


6  Arithmetic. 

7.  A  number  dictated,  or  enunciated,  is  written  then,  by  plac- 
ing one  after  the  other,  beginning  at  the  left,  the  figures  wbich 
express  the  number  of  units  of  each  collection  5  but  it  is  neces- 
sary to  keep  in  mind  the  order  in  which  the  collections  succeed 
each  other,  that  no  one  may  be  omitted,  and  to  put  ciphers  in  the 
room  of  those,  whicb  are  wanting  in  the  enunciation  of  the  num- 
ber to  be  written.  If,  for  example,  the  number  were  three  hun- 
dred and  twenty  four  thousand^  nine  hundred  and  four,  we  should 
put  3  for  the  hundreds  of  thousands,  2  for  the  twenty  thousand, 
or  the  two  tens  of  thousands,  4  for  the  thousands,  9  for  the  hun- 
dreds J  and  as  the  tens  come  immediately  after  the  hundreds, 
and  are  wanting  in  the  given  number,  we  should  put  a  cipher  in 
the  room  of  them,  and  then  write  the  figure  4  for  the  units  ;  we 
should  thus  have  324904. 

In  the  same  way,  writing  ciphers  in  the  place  of  tens  of  thou- 
sands, thousands  and  tens,  which  are  wanting  in  the  number  five 
hundred  thousand  three  hundred  and  two,  we  should  have  500302. 

8.  When  a  number  is  written  in  figures,  in  enunciating  it,  or 
expressing  it  in  language,  it  is  necessary  to  substitute  for  each 
of  the  figures  the  word  which  it  represents,  and  then  to  mention 
the  collection  of  units,  to  whiclf  it  belongs  according  to  the  place 
it  occupies.     The  following  example  will  illustrate  this ; 


4, 

8 

9 

"^J 

3 

2 

1, 

5 

8 

0, 

3 

4 

6, 

g 

c 
a 

H 

i 

1 

2 

a 

c 

H 

w 

c 
s 

^ 

s 

d 

!zi 

1 

0 

0 

1 

0 
2 

1 

en 

1 

2, 

Ui 

5 

S 

cc 

0 

0 

H^ 

0 

0 

«» 

J—; 

•-^ 

f^ 

Ss 

0 

te 

,  0' 

3 

0 

g 

s 

«! 

S 

1 

The  figures  of  this  number  are  divided  by  commas,  into  portions 
of  three  figures  each,  beginning  at  the  right ;  but  the  last  divis- 
ion on  the  left,  which  in  the  present  instance  has  but  two  figures, 
may  sometimes  have  but  one.  Each  of  these  divisions  corres- 
ponds to  the  collections  designated  by  the  words  7init,  thousand, 


JtddiUon,  7 

miUionf  hillion,  irUliorif  and  their  figures  express  successively 
the  units,  tens  and  hundreds  of  each.  Consequently f  the  expression 
of  the  'whole  number  given  is  made  in  words,  hy  reading  each  divis- 
ion of  jigures  as  if  it  stood  alone,  and  adding,  after  its  units,  the 
name  of  their  place. 

The  above  example  is  read,  twenty  four  trillions,  eight  hundred 
and  ninety  seven  billions,  three  hundred  and  twenty  one  millio7is. 
Jive  hundred  and  eighty  thousand,  three  hundred  and  forty  six  units. 

9.  Numbers  admit  of  being  considered  in  two  ways  ;  one  is, 
when  no  particular  denomination  is  mentioned,  to  which  their 
units  belong,  and  they  are  then  called  abstract  numbers;  the 
other  when  the  denomination  of  their  units  is  specified,  as  when 
we  say,  two  men,  five  years,  three  hours,  &c.  tliese  are  called 
concrete  numbers. 

It  is  evident,  that  the  formation  of  numbers,  by  the  successive 
union  of  units,  is  independent  of  the  nature  of  these  «nits,  and 
that  this  must  also  be  the  case  with  the  properties  resulting  from 
this  formation  ;  by  which  properties  we  are  enabled  to  compound 
and  decompound  numbers,  which  is  called  calculation.  We  shall 
now  explain  the  principal  rules  for  the  calculation  of  numbers, 
■without  regai'd  to  the  nature  of  their  units. 

Addition. 

10.  This  operation,  which  has  for  its  object  the  uniting  of 
several  numbers  in  one,  is  only  an  abbreviation  of  the  formation 
of  numbers  by  the  successive  union  of  units.  If,  for  instance,  it 
were  required  to  add  five  to  seven,  it  would  be  necessary,  in  the 
series  of  the  names  of  numbers,  on£,  two,  three,  four,  fve,  six, 
seven,  &c.  to  ascend  five  places  above  seven,  and  we  should  then 
come  to  the  word  twelve,  which  is  consequently  the  amount  of 
seven  units  added  to  five.  It  is  upon  this  process  that  the  ad- 
dition of  all  small  numbers  depends,  the  results  of  which  are 
committed  to  memory;  its  immediate  application  to  larger  num- 
bers would  be  impossible,  but  in  this  case,  we  suppose  these 
numbers  divided  into  the  different  collections  of  units  contained 
in  them,  and  we  may  add  together  those  of  the  same  name.  For 
instance,  to  add  27  to  32,  we  add  the  7  units  of  tlic  first  number 
to  the  2  of  the  second,  making  9  ;  then  the  2  tens  of  the  first  with 


S  Jiritfimetic. 

the  3  of  the  second,  making  5  tens.  The  two  results,  taken  to- 
gether, form  a  total  of  5  tens  and  9  units  or  59,  which  is  the  sum 
of  the  numbers  proposed. 

What  is  here  said,  applies  to  all  numbers  however  large,  that 
are  to  be  added  together,  but  it  is  necessary  to  observe  that  the 
partial  sums,  resulting  from  the  addition  of  two  numbers,  each 
expressed  by  a  single  figure,  often  contain  tens,  or  units  of  the 
next  higher  collection,  and  these  ought  consequently  to  be  joined 
to  their  proper  collection. 

In  the  addition  of  the  numbers  49  and  78,  the  sum  of  the  units  9 
and  8  is  17,  of  which  we  should  reserve  10,  or  ten,  to  be  added  to 
the  sum  of  the  tens  in  the  given  numbers  ;  next  we  say  that  4  and 
7  make  11,  and  joining  to  this  the  ten  we  reserved,  we  have  12 
for  the  number  of  tens  contained  in  the  sum  of  the  given  num- 
bers J  which  sum,  therefore,  contains  1  hundred,  2  tens  and  T 
units,  that  is  127. 

1 1.  By  proceeding  on  these  principles,  a  method  has  been  devis- 
de  of  placing  numbers,  that  are  to  be  added,  which  facilitates  the 
uniting  of  tiieir  collections  of  units,  and  a  rule  has  been  formed 
which  the  following  example  will  illustrate. 

Let  the  numbers  be  527,  2519,  9812,  73  and  8 ;  in  order  to 
add  them  together,  we  begin  by  writing  them  under  each  other, 
placing  the  units  of  the  same  order  in  the  same  column ;  then 
we  draw  a  line  to  separate  them  from  the  result,  which  is  to  be 
written  undei-neath  it. 

527 

2519 

9812 

73 

8 

Sum  12939 
We  at  first  find  the  sum  of  the  numbers  contained  in  the  column 
of  units  to  be  29,  we  write  down  only  the  nine  units,  and  reserve 
the  2  tens,  to  be  joined  to  those  which  are  contained  in  the  next  col- 
umn, which,  thus  increased,  contains  13  units  of  its  own  order; 
we  write  down  here  only  the  three  units,  and  carry  the  ten  to 
the   next  column.    Proceeding  with  this  column  as  with  the 


Subtraction.  9 

others,  we  find  its  sum  to  be  19 ;  we  write  down  the  9  units  and 
carry  the  ten  to  the  next  column,  the  sum  of  which  we  then  find 
to  be  12;  we  write  down  the  2  units  under  this  cohnnn  and 
place  the  ten  on  the  left  of  it ;  that  is,  we  write  down  the  sum  of 
this  column,  as  it  is  found. 

By  this  means  we  obtain  12939  for  the  sum  of  the  given  num- 
bers. 

12.  The  rule  for  performing  this  operation  may  be  given  thus. 

Write  the  numbers  to  be  added,  under  each  other,  so  that  all  the 
units  of  the  same  kind  maij  stand  in  the  same  column,  and  draw  a 
line  under  them. 

Beginning  at  the  right,  add  up  successively  the  numbers  in  each 
column;  if  the  sum  does  iwt  exceed  9,  write  it  beneath  its  column, 
as  it  is  found;  if  it  contains  one  or  more  tens,  carry  them  to  the 
next  column ;  lastly,  under  the  last  column  write  the  whole  of  its 
sumf. 

Examples  for  practice. 

Add  together  86S5,  2194,  7421,  S063,  2196  and  1225. 

Ans,  26734. 
Add  together  84371,  6250,  10,  3842  and  631.  Ms.  95104. 
Add  together  3004,  523,  8710,  6345  and  784.  dns,  19366. 
Add  together  7861,  345,  8023.  Jm.  16229. 

Add  together  66947,  46742  and  132684.  Ans.  246373. 

f 

Subtraction. 

13.  After  having  learned  to  compose  a  number  by  the  addi- 
tion of  several  others,  the  first  question,  that  presents  itself,  is, 
how  to  take  one  number  from  another  that  is  greater,  or  which 
amounts  to  the  same  thing,  to  separate  this  last  into  two  parts,  one 
of  which  shall  be  the  given  number.  If,  for  instance,  we  have  the 

t  The  best  method  of  proving  addition  is  by  means  of  subtraction 
The  learner  may  however,  in  general,  satisfy  himself  of  the  correct 
ness  of  his  work  by  beginning  at  the  top  of  each  column  and  adding 
down,  or  by  separating  the  upper  line  of  figures  and  adding  up  the 
rest  and  then  adding  this  sum  to  the  upper  line. 


10  Arithmetic. 

number  9,  and  we  wish  to  take  4  from  it,  we  should,  by  doing 
this,  separate  it  into  two  parts,  which  by  addition  would  be  the 
same  again. 

To  take  one  number  from  another,  when  they  are  not  large, 
it  is  necessary  to  pursue  a  course  opposite  to  that  prescribed,  in 
the  beginning  of  article  10,  for  finding  their  sum ;  that  is,  in 
the  series  of  the  names  of  numbers,  we  ought  to  begin  from  the 
greatest  of  the  numbers  in  question,  and  descend  as  many  places 
as  there  are  units  in  the  smallest,  and  we  shall  come  to  the  name 
given  to  the  difference  required.  Thus,  in  descending  four 
places  below  the  number  nine^  we  come  to  jive,  which  expresses 
the  number  that  must  be  added  to  4  to  make  9,  or  which  shows 
how  much  9  is  greater  than  4. 

In  this  last  point  of  view,  5  is  the  excess  of  9  above  4.  If  we 
only  wished  to  show  the  inequality  of  the  numbers  9  and  4,  with- 
out fixing  our  attention  on  the  order  of  their  values,  we  should 
say  that  their  difference  was  5.  Lastly,  if  we  were  to  go  through 
the  operation  of  taking  4  from  9,  we  should  say  that  the  re- 
mainder is  5.  Thus  we  see  that,  although  the  words,  excess, 
remainder,  and  difference,  are  synonymous,  each  answers  to  a 
particular  manner  of  considering  the  separation  of  the  number  9 
into  the  parts  4  and  5,  winch  operation  is  always  designated  by 
the  name  subtraction. 

14.  When  the  numbers  are  large,  the  subtraction  is  perform- 
ed, part  at  a  time,  by  taking  successively  from  the  units  of  each 
order  in  the  greatest  number,  the  corresponding  units  in  the 
least.  That  this  may  be  done  conveniently,  the  numbers  are 
placed  as  9587  and  345  in  the  following  example^ 
9587 
S45 

Hemainder    9242 
and  under  each  column  is  placed  the  excess  of  the  upper  number, 
in  that  column,  over  the  lower,  thus ; 

5,  taken  from  7,  leaves  2, 

4,  taken  trom  8,  leaves  4, 

3,  taken  from  5,  leaves  2, 
and  writing  afterwards  the  figure  9,  from  which  there  is  noth- 


Subtraction,  11 

ing  to  betaken ;  the  remainder,  9242,  shows  how  much  95B7  is 
greater  than  345. 

That  the  process  here  pursued  gives  a  true  result  is  indispu- 
table, because  in  taking  from  the  greatest  of  the  two  numbers 
all  the  parts  of  the  least,  we  evidently  take  from  it  the  whole  of 
the  least. 

15.  The  application  of  this  process  requires  particular  atten- 
tion, when  some  of  the  orders  of  units  in  the  upper  number  are 
greater  than  the  corresponding  orders  in  the  lower. 

If,  for  instance,  397  is  to  be  taken  from  524. 
524 
397 

Remainder  1 27 
In  performing  this  question  we  cannot  at  first  take  the  units 
in  the  lower  number  from  those  in  the  upper ;  but  the  number 
524,  here  represented  by  4  units,  2  tens  and  5  hundreds,  can  be 
expressed  in  a  different  manner  by  decomposing  some  of  its  col- 
lections of  units,  and  uniting  a  part  with  the  units  of  a  lower 
order.  Instead  of  the  2  tens  and  4  units  which  terminate  it  we 
can  substitute  in  our  minds  1  ten  and  14  units,  then  taking  from 
these  units  the  7  of  the  lower  number,  we  get  the  remainder  7. 
By  this  decomposition,  the  upper  number  now  has  but  one  ten, 
from  which  we  cannot  take  the  9  of  the  lower  number,  but  from 
the  5  hundred  of  the  upper  number  we  can  take  1,  to  join  with 
the  ten  that  is  left,  and  we  shall  then  have  4  hundreds  and  11 
tens,  taking  from  these  tens  the  tens  of  the  lower  number,  2  will 
remain.  Lastly,  taking  from  the  4  hundreds,  that  are  left  in 
the  upper  number,  the  three  hundreds  of  the  lower,  we  obtain  the 
remainder  1,  and  thus  get  127  as  the  result  of  the  operation. 

This  manner  of  working  consists,  as  we  sec,  in  borrowing, 
from  the  next  higher  order,  an  unit,  and  joining  it  according  to 
its  value  to  those  of  the  order,  on  which  we  are  employed,  ob- 
serving to  count  the  upper  figure  of  the  order  from  which  it  was 
borrowed  one  unit  less,  when  we  shall  have  come  to  it. 

16.  When  any  orders  of  units  are  wanting  in  the  upper  num- 
ber, that  IS,  when  there  are  ciphers  between  its  figures,  it  is 


is  Arithmetic. 

necessary  to  go  to  the  first  figure  on  the  left,  to  borrow  the  10  that 
is  wanted.     See  an  example 

7002 
3495 

Remainder  3507. 

As  we  cannot  take  the  5  units  of  the  lower  number  from  the  2 
of  the  upper,  we  borrow  10  units  from  the  7000,  denoted  by  the 
figure  7,  which  leaves  6990  ;  joining  the  10  we  borrowed  to  the 
figure  2.  the  upper  number  is  now  decompounded  into  6990  and 
12  ;  taking  from  12  the  5  units  of  the  lower  number,  we  obtain 
7  for  the  units  of  the  remainder. 

This  first  operation  has  left  in  the  upper  number  6990  units 
or  699  tens  instead  of  the  700  expressed  by  the  three  last  figures 
on  the  left ;  thus  the  places  of  the  two  ciphers  are  occupied  by 
9s  and  the  significant  figure  on  the  left  is  diminished  by  unity. 
Continuing  the  subtraction  in  the  other  columns  in  the  same 
manner,  no  difficulty  occurs,  and  we  find  the  remainder,  as  put 
down  in  the  example. 

17.  Recapitulating  the  remarks  made  in  the  two  preceding 
articles,  the  rule  to  be  obsei  ved  in  performing  subtraction  may 
be  given  thus.  Plcuie  the  less  number  under  the  greater,  so  that 
their  units  of  the  same  order  may  be  in  the  same  column,  and  draw 
a  line  under  them ;  beginning  at  the  right  take  successively  each 

figure  of  the  lower  number  from  the  one  in  the  same  column  of  the 
upper  ;  if  this  cannot  be  done,  increase  the  upper  figure  by  ten  unitSf 
counting  the  next  significant  figure,  in  the  upper  member,  less  by 
unity,  and  if  ciphers  come  between,  regard  them  as  9s. 

18.  For  greater  convenience,  when  it  is  necessary  to  decrease 
the  upper  figure  by  unity,  we  can  suffer  it  to  retain  its  value, 
and  add  this  unit  to  the  corresponding  lower  figure,  which,  thus 
increased,  gives  as  is  wanted,  a  result  one  less  than  would  arise 
from  Ihe  written  figures.  In  the  first  of  the  following  examples, 
after  having  taken  6  units  from  14,  we  count  the  next  figure  of 
the  lower  number  8,  as  9,  and  so  in  the  others. 


Multiplication. 

Examples, 

16844 

10378 

103034 

49812002 

9786 

2437 

69845 

18924983 

7058 

33189 

173425 

8037142 

2123724 • 

39742107 

57632 

5067310 

1123467 

25378421 

13 


Method  of  proving  Addition  and  Suhiradion, 

19.  In  performing  an  operation,  according  to  a  process,  the 
correctness  of  wiiich  is  established  upon  fixed  principles,  we  may 
nevertheless  sometimes  commit  errours  in  the  partial  additions 
and  subtractions,  the  results  of  which  we  seek  in  the  memory. 
To  prevent  any  mistake  of  this  kind,  we  have  recourse  to  a  me- 
thod, the  reverse  of  the  first  operation,  by  which  we  ascertain 
whether  the  results  are  right ;  this  is  called  proving  the  operation. 

The  proof  of  addition  consists  in  subtracting  successively  from 
the  sum  of  the  numbers  added,  all  the  parts  of  these  numbers,  and 
if  the  work  has  been  correctly  performed,  there  wUl  be  no  re- 
mainder. ,  We  will  now  show  by  the  example  given  in  article  11, 
how  to  perform  all  these  subtractions  at  once. 
527 
2519 
9812 

8 


Sum 


12939 
1120 


We  first  add  the  numbers  in  the  left  hand  column,  which 
here  contains  thousands,  and  subtract  the  sum  11  from  12, 
which  begins  the  preceding  result,  and  write  underneath  the 
diflFerence  1,  produced  by  what  was  reserved  from  the  column 
of  hundreds,  in  performing  the  addition.  The  sum  of  the 
column  of  hundreds  taken  by  itself,  amounts  to  but  18,  if  we  take 


14  ^irithmetic. 

this  from  the  9  of  the  first  result,  increased  by  borrowing  the  one 
thousand,  considered  as  ten  hundred,  that  remains  from  the 
column  preceding  it  on  the  left,  the  remainder  1,  written  beneath 
will  show  what  was  reserved  from  the  column  of  tens.  The  sum 
of  these  last  1 1,  taken  from  13,  leaves  for  its  remainder  2  tens, 
the  number  reserved  from  the  column  of  unifs.  Joining  these 
2  tens  with  the  9  units  of  the  answer,  we  form  the  number  29, 
which  ought  to  be  exactly  the  sum  of  the  column  of  units,  as  this 
column  is  not  affected  by  any  of  tlie  others ;  adding  again  the 
numbers  in  this  column,  we  ought  to  come  to  the  same  result,  and 
consequently,  to  have  no  rcnmitjder.  This  is  actually  the  case, 
as  is  denoted  by  the  0  written  under  the  column.  The  process, 
just  explained,  may  be  given  thus  ;  to  provte  addition^  beginning 
on  the  left,  add  again  each  of  the  several  columns,  subtract  the  sums 
respectively  from  the  sums  written  above  them  and  write  down  the 
remainders,  which  must  be  joined,  each  as  so  many  tens  to  the 
sum  of  the  next  column  on  the  right ;  if  the  work  be  correct  there  wiU 
be  no  remainder  under  the  last  column. 

20.  The  proof  of  subtraction  is,  that  the  remainder,  added  to 
the  least  number,  exactly  gives  the  greatest.  Thus  to  ascertain 
the  exactness  of  the  following  subtraction, 

524 

297 


524 
we  add  the  remainder  to  the  smallest  number,  and  find  the  sum, 
in  reality,  equal  to  the  greatest. 

Multiplication.   . 

21.  When  the  numbers  to  be  added  are  equal  to  each  other, 
addition  takes  the  name  of  multiplication,  because  in  this  case  the 
sum  is  composed  of  one  of  the  numbers  repeated  as  many  times 
as  tliere  are  numbers  to  be  added.  Reciprocally,  if  we  wish  to 
repeat  a  number  several  times,  we  may  do  it,  by  adding  the  num- 
ber to  itself  as  many  times,  wanting  one,  as  it  is  to  be  repeated. 
For  instance,  by  the  following  addition, 


Multiplication.  IS 

16 
16 
16 
16 

64 
the  number  16  is  repeated  four  times,  and  added  to  itself  three 
times. 

To  repeat  a  number  twice  is  to  double  it ;  3  times,  to  triple  it  j 
4  times,  to  quadruple  it,  and  so  on. 

22.  Multiplication  implies  three  numbers,  namely,  that,  which 
is  to  be  repeated,  and  which  is  called  the  multiplicand  ;  the  num- 
ber which  shows  how  many  times  it  is  to  be  rejjeated,  which  is 
called  the  multiplier ;  and  lastly,  the  result  of  the  operation, 
which  is  called  the  product.  The  midtiplicand  and  multiplier, 
considered  as  concurring  to  form  the  product,  are  called  factors 
of  the  product*  In  the  example  given  above,  16  is  the  multipli- 
candf  4  the  multiplier,  and  64  the  product ;  and  we  see  tliat  4  and 
16  are  i\w,  factors  of  64. 

23.  When  the  multiplicand  and  multiplier  are  large  numbers, 
the  formation  of  the  product,  by  the  repeated  addition  of  the 
multiplicand,  would  be  very  tedious.  In  consequence  of  this, 
means  have  been  sought  of  abridging  it,  by  sei)atating  it  into  a 
certain  number  of  partial  operations,  easily  performed  by  mem- 
ory. B^or  instance,  the  number  16  would  be  repeated  4  times, 
by  taking  separately,  the  same  number  of  times,  the  6  units  and 
the  ten,  that  compose  it.  It  is  sufficient  then  to  know  the  pro- 
ducts arising  from  the  multiplication  of  the  units  of  each  order 
in  the  multiplicand  by  the  multiplier,  when  the  multiplier  con- 
sists of  a  sljigle  figure,  and  this  amounts,  for  all  cases  that  can 
occur,  to  finding  the  products  of  each  one  of  the  9  first  numbers 
by  every  other  of  these  numbers. 

24.  These  products  are  contained  in  the  following  table,  attri- 
buted to  Pythagoras. 


le 


»inthmetic. 


TABLE  OF  PYTHAGORAS. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

8  ^ 

12 

16 

§0 

24 

28 

32 

36 

5 

10 

15 

20 

25 

30 

35 

40 

48 

45 

6 

12 

18 
21 

24 

24 

30 

36 

42 

54 

7 

14 

28 

35 

42 

49 

66 

63 

8 

16 

32 

40 

48 

56 

64 

72 

9 

18 

27 

36 

45 

54 

63 

72 

81 

25.  To  form  this  table,  the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  9, 
are  written  first  on  the  same  line.  Each  one  of  these  numbers 
is  then  added  to  itself  and  the  sum  written  in  the  second  line, 
which  thus  contains  each  number  of  the  first  doubled,  or  the 
product  of  each  number  by  2.  Each  number  of  the  second  line 
is  then  added  to  the  number  over  it  in  the  first,  and  their  sums 
are  written  in  the  third  line,  which  thus  contains  the  triple  of 
each  number  in  tlic  first,  or  their  products  be  3.  By  adding  the 
numbers  of  the  third  line  to  those  of  the  first,  a  fourth  is  formed, 
containing  tlie  quadruple  of  each  number  of  the  first,  or  their 
products  by  4  ;  and  so  on,  to  the  ninth  line,  which  contains  the 
products  of  each  number  of  the  first  line  by  9. 

It  may  not  be  amiss  to  remark,  that  the  different  products  of 
any  number  whatever  by  the  numbers  2,  3,  4,  5,  &c.  are  called 
multiples  of  that  number ;  thus  6,  9,  12,  15,  &;c.  are  multiples  of  3. 

26.  When  the  formation  of  this  table  is  well  understood,  the 
mode  of  using  it  may  be  easily  conceived.  If,  for  instance,  the 
product  of  7  by  5  were  required ;  looking  to  the  fiftli  line,  which 
contains  the  different  products  of  the  9  first  numbers  by  5,  we 
should  take  the  one  directly  under  the  7,  which  is  35  ;  the  same 


Multipli6ation.  If 

method  should  be  pursued  in  every  other  instance,  and  the  pro- 
duct wUl  always  be  found  in  the  line  of  the  multiplier  and  under 
the  multiplicand, 

27.  If  we  seek  in  the  table  of  Pythagoras  the  product  of  5  by 
7,  we  shall  find,  as  before,  35,  although  in  this  case  5  is  the  inul- 
plicand,  and  7  the  multiplier.  This  remark  is  applicable  to  ^ach 
product  in  the  table,  and  it  is  possible,  in  any  multiplication^  to 
reverse  the  order  of  the  factors  ;  that  is,  to  make  the  multiplicand 
the  multiplier,  and  the  multiplier  the  multiplicand. 

As  the  table  of  Pythagoras  contains  but  a  limited  number  of 
products,  it  would  not  be  sufficient  to  verify  the  above  conclu- 
sion, by  this  table ;  for  a  doubt  might  arise  respecting  it  in  the 
case  of  greater  products,  the  number  of  which  is  unlimited ; 
there  is  but  one  method  independent  of  the  particular  value  of 
the  multiplicand  and  multiplier,  of  showing  that  there  is  no  ex- 
ception to  this  remark.  This  is  one  well  calculated  for  the  pur- 
pose, as  it  gives  a  good  illustration  of  the  manner,  in  which  the 
product  of  two  numbers  is  formed.  To  make  it  more  easily  un- 
derstood, we  will  apply  it  first  to  the  factors  5  and  3. 

If  we  write  the  figure  1,  5  times  on  one  line,  and  place  two 
similar  lines  underneath  the  first,  in  this  manner, 

1,  1,  1,  1,  1, 

1,  1,  1,  1,  1, 

1,  1,  1,  1,  1, 
the  whole  number  of  Is  will  consist  of  as  many  times  5  as  there  are 
lines,  that  is,  3  times  5 ;  but,  by  the  disposition  of  tliese  lines,  the 
figures  are  ranged  in  columns,  containing  3  each.  Counting  them 
in  this  manner,  we  find  as  many  times  3  units  as  there  are  col- 
umns, or  5  times  3  units,  and  as  the  product  does  not  depend  on 
the  manner  of  counting,  it  follows  that  3  times  5,  and  5  times  3 
give  the  same  product.  It  is  easy  to  extend  this  reasoning  to 
any  numbers,  if  we  conceive  each  line  to  contain  as  many  units 
as  there  are  in  the  multiplicand,  and  the  number  of  lines,  plac- 
ed one  under  the  other,  to  be  equal  to  the  multiplier.  In  count- 
ing the  product  by  lines,  it  arises  from  the  multiplicand  repeated 
as  many  times  as  there  are  units  in  the  multiplier ;  but  the  as- 
semblage of  figures  written,  presents  as  many  columns  as  there 


18  Arithmetic. 

are  units  in  a  line,  and  each  column  contains  as  many  units  as 
there  are  lines ;  if  then,  we  choose  to  count  by  columns,  the 
number  of  lines,  or  the  multiplier,  will  be  repeated  as  many 
times  as  there  are  units  in  a  line,  that  is,  in  the  multiplicand.  We 
may  therefore,  in  finding  the  product  of  any  two  numbers,  take 
either  of  tliem  at  pleasure,  for  the  multiplier. 

28.  The  reasoning  just  given  to  prove  the  truth  of  the  pre- 
ceding proposition,  is  the  demonstration  of  it,  and  it  may  be 
remarked,  that  the  essential  distinction  of  pure  mathematics  is> 
that  no  proposition,  or  process,  is  admitted,  which  is  not  th« 
necessary  consequence  of  the  primary  notions,  on  which  it  is 
founded,  or  the  truth  of  which  is  not  generally  established  by- 
reasoning  independent  of  particular  examples,  which  can  never 
constitute  a  proof,  but  serve  only  to  facilitate  the  readei^'s  under- 
standing tlie  reasoning,  or  the  practice  of  the  rules. 

29.  Knowing  all  the  products  given  by  the  nine  first  numbers, 
combined  with  each  other,  we  can,  according  to  the  remark  in 
article  23,  multiply  any  number  by  a  number  consisting  of  a 
single  figure,  by  forming  successively  the  product  of  each  order 
of  units  in  the  multiplicand,  by  the  multiplier ;  the  work  is  as 
follows ; 

526 
7 

3682 

The  product  of  the  units  of  the  multiplicand,  6,  by  the  multi- 
plier, 7,  being  42,  we  write  down  only  the  2  units,  reserving  the 
4  tens  to  be  joined  with  those,  that  will  be  found  in  the  next 
higher  place. 

The  product  of  the  tens  of  the  multiplicand,  2,  by  the  multi- 
plier, 7,  is  14,  and  adding  the  4  tens  we  reserved,  we  make  them 
18,  of  which  number  we  write  only  the  units,  and  reserve  the 
ten  for  the  next  operation. 

The  product  of  the  hundreds  of  the  multiplicand,  5,  by  the 
multiplier,  7,  is  35,  wlien  increased  by  the  1  we  reserved,  it  be- 
comes 36,  the  whole  of  which  is  written,  because  there  are  no 
more  figures  in  the  multiplicand. 

30.  This  process  may  be  given  thus  ;    To  multiply  a  numhei" 


I       .         '  Multiplication.  19 

of  several  Jigiires  hij  a  single  figure,  place  the  multiplier  under  the 
units  of  the  multiplicand^  and  draw  a  line  heneath,  to  separate  them 
from  the  p-oduct.  Beginning  at  the  rights  multiply  successively,  hy 
the  multiplier,  the  units  of  each  order  in  the  midtiplicand,  and 
ivrite  the  whole  product  of  each,  when  itdoes  not  exceed  9  ;  but  if 
it  contains  tens,  reserve  them  to  be  added  to  the  next  product.  Con- 
tinue thus  to  the  last  figure  of  the  multiplicand,  on  the  left,  the 
whole  residt  of  which  must  be  written  down. 

Examples.  243  by  6.  Ans.  1458.     8943  by  9.    Ans.  80487. 

It  is  evident  that,  when  the  multiplicand  is  terminated  by  Os, 
the  operation  can  commence  only  with  its  first  significant  figure ; 
but  to  give  the  product  its  proper  value,  it  is  necessary  to  put, 
on  the  right  of  it,  as  many  Os  as  there  are  in  the  multiplicand. 
As  for  the  Qs,  which  may  occur  between  the  figures  of  the  mul- 
tiplicand, they  give  no  product,  and  a  0  must  be  written  down 
when  no  number  has  been  reserved  from  the  preceding  product, 
as  is  shown  by  the  following  examples ; 

956  8200  7012  80970 

6  9  5  4 


5736  73800  35060  323880 

Multiply 

730  by  3.  Ans.  2190.  8104  by  4.  Ms,  32416. 

20508  by  5.  Ans.  102540.        360500  by  6.  Ans.  2163000. 

297000  by  7.  Ans.  2079000.  9097030  by  9.  Ans.  81873270. 

31.  The  most  simple  number,  expressed  by  several  figures, 
being  10,  100,  1000,  &c.  it  seems  necessary  to  inquire  how  we 
can  multiply  any  number  by  one  of  these.  Now  if  we  recollect 
the  principle  mentioned  in  article  6,  by  which  the  same  figure  is 
increased  in  value  10  times,  by  every  remove  towards  the  left, 
we  shall  soon  perceive,  that,  to  multiply  any  number  by  10,  we 
must  make  each  of  its  order.?  of  units  ten  times  greater ;  that 
is,  we  must  change  its  units  into  tens,  its  tens  into  hundreds, 
and  so  on,  and  that  this  is  effected  by  placing  a  0  on  the  right 
of  the  number  proposed,  because  then  all  its  significant  figures 
will  be  advanced  one  place  towards  the  left. 

For  the  same  reason,  to  multiply  any  number  by  100,  we 
should  place  two  ciphers  on  the  right ;  for,  since  it  becomes  ten 


20  ^rithmetie. 

times  greater  by  the  first  cipher,  the  second  will  make  it  ten 
times  greater  still,  and  consequently  it  will  be  10  times  10,  or 
100  times,  greater  than  it  was  at  first. 

Continuing  this  reasoning,  it  will  be  perceived  that,  accord- 
ing to  our  system  of  numeration,  a  number  is  multiplied  by  10, 
100, 1000,  &c.  by  writing  on  the  right  of  the  multiplicand  as 
many  ciphers,  as  there  are  on  the  right  of  the  unit  in  the  multi- 
plier. 

32.  When  the  significant  figure  of  the  multiplier  differs  from 
unity,  as,  for  instance,  when  it  is  required  to  multiply  by  30,  or 
300,  or  3000,  which  are  only  10  times  3,  or  100  times  3,  or  1000 
times  3,  &c.  the  operation  is  made  to  consist  of  two  parts,  we  at 
•first  multiply  by  the  significant  figure,  3,  according  to  the  rule 
in  article  30,  and  then  multiply  the  product  by  10,  100,  or  1000, 
kc.  (as  was  stated  in  the  preceding  article,)  by  WTiting  one,  two, 
three,  &c,  ciphers  on  the  right  of  this  product. 

Let  it  be  required,  for  instance,  to  multiply  764  by  300. 
764 
300 

229^00 

The  four  significant  figures  of  this  product  result  from  the 
multiplication  of  764  by  3,  and  are  placed  two  places  towards 
the  left  to  admit  the  two  ciphers,  which  terminate  the  multiplier. 

In  general,  when  the  Tmdtiplier  is  terminated  by  a  number  of 
ciphers,  Jirst  multiply  the  multiplicand  by  the  significant  Jigure  of 
the  multiplier,  and  place,  after  the  product,  as  many  dphers  as  there 
are  in  the  multiplier. 

Examples, 

Multiply 
35012  by  100.  Ms.  3501200.       638427  by  500.  Ms.  31921 35Q0. 
2107900  by  70.  ./Zns.147553000.  9120400  hy  90.  Ms.  820836000. 

S3.  The  preceding  rules  apply  to  the  case,  in  which  the  multi- 
plier is  any  number  whatever,  by  considering  separately  each 
of  the  collections  of  units  of  which  it  is  composed.  To  multiply, 
for  instance,  793  by  345,  or  which  is  the  same  thing,  to  repeat 
793, 345  times,  is  to  take  793,  5  times,  added  to  40  times,  added  to 


Multiplication.  21 

300  times,  and  the  operation  to  be  performed  is  resolved  into  3 
others,  in  each  of  which  the  multipliers,  5,  40,  and  300,  have  but 
one  significant  figure. 

To  add  the  result  of  these  three  operations  easily,  the  calcula- 
tion is  disposed  thus  j 

793 

345 

3965 

31720 

237900 

273585 

The  multiplicand  is  multiplied  successively  by  the  units,  tens, 
hundreds,  &c.  of  the  multiplier,  observing  to  place  a  cipher  on 
the  right  of  the  partial  product,  given  by  the  tens  in  the  multi- 
plier, and  two  on  the  right  of  tUe  product  given  by  hundreds, 
which  advances  the  first  of  these  products  one  place  towards  the 
left,  and  the  second,  two.  The  three  partial  products  are  then 
added  together,  to  obtain  the  total  product  of  the  given  numbers. 

As  the  ciphers,  placed  at  the  end  of  tiiese  partial  products,  are 
of  no  value  in  the  addition,  we  may  dispense  w  ith  writing  them, 
provided  we  take  care  to  put  in  its  proper  place,  the  first  figure 
of  the  product  given  by  each  significant  figure  of  the  multiplier ; 
that  is,  to  put  in  the  place  of  tens,  the  first  figure  of  the  product 
given  by  the  tens  in  the  multiplier ;  in  the  place  of  hundreds  the 
first  figure  of  the  product  given  by  the  hundreds  in  the  multiplier, 
and  so  on. 

34.  According  to  what  has  been  said,  the  rule  is  as  follows. 
To  multiply  any  two  numbers^  one  hy  the  other,  form  successively 
Caccording  to  the  rule  in  article  30,  J  the  products  of  the  imdtipli~ 
cand,  by  the  different  orders  of  units  in  the  rmiltiplier  ;  observing  ta 
place  the  Jirst  figure  of  each  partial  product  under  the  units  of  the 
same  order  with  thefgure  of  the  multiplier,  by  which  the  product  is 
given;  and  tJien  add  together  all  the  partial  products. 

35.  When  the  multiplicand  is  terminated  by  ciphers,  they  may 
at  first  be  neglected,  and  all  the  partial  multiplications  begin 
with  the  first  significant  figure  of  the  multiplicand ;  but  after- 


^S  .Arithmetic. 

wards,  to  put  in  their  proper  rank  the  figures  of  the  total  pro- 
duct, as  many  ciphers,  as  thei-e  are  in  the  multiplicand,  must  he 
written  on  tlie  right  of  this  product. 

If  the  multiplier  is  terminated  by  ciphers,  we  may,  according 
to  the  remark  in  article  31,  neglect  these  also,  provided  we  write 
an  equal  number  on  the  right  of  the  product. 

Hence  it  results  that,  when  both  multiplicand  and  multiplier  are 
terminated  by  ciphers,  these  ciphers  may  at  Jirst  be  neglected,  and 
after  the  other  Jigures  of  the  product  are  obtained,  the  same  number 
may  be  written  on  the  right  of  the  product. 

When  there  are  ciphers  between  the  significant  figures  of  the 
multiplier,  as  they  give  no  product,  they  may  be  passed  over, 
observing  to  put  in  its  proper  place,  the  unit  of  the  product  given 
by  the  figure  on  the  left  of  these  ciphers. 

Examples. 

500  526     Multiply  9648  by  5137.  ^ns.  49561776. 

40  307  7854  by  350.  Ans.  27489000. 

17204774  by  125.  Jliis.  2150596750. 


12000  3682  62500  by  520.   .5?is.  32500000. 

157800  25980762  by  20.  ^ns.  10392304800. 

161482 

Division. 

36.  The  product  of  two  numbers  being  formed  by  repeating  one 
of  these  numbers  as  many  times  as  there  are  units  in  the  other, 
we  can,  from  the  product,  find  one  of  the  factors,  by  ascertaining 
how  many  times  it  contains  the  other ;  subtraction  alone  is  neces- 
sary for  this.  Thus,  if  it  be  required  to  ascertain  the  number 
of  times  64  contains  16,  we  need  only  subtract  16  from  64  as 
many  times  as  it  can  be  done ;  and  since,  after  4  subtractions, 
nothing  is  left,  we  conclude,  that  16  is  contained  4  times  in  64. 
This  manner  of  decomposing  one  luimber  by  another,  in  order 
to  know  how  many  times  the  last  is  contained  in  the  first,  is 
called  division,  because  it  serves  to  divide,  or  portion  out,  a 
given  number  into  equal  parts,  of  which  the  number  or  v  alue  is 
given* 


Divisim,  23 

If,  for  instance,  it  were  required  to  divide  64  into  4  equal 
parts ;  to  find  the  value  of  these  parts,  it  would  be  necessary  to 
ascertain  the  number,  that  is  contained  4  times  in  64,  and  conse- 
quently to  regard  64  as  a  product,  having  for  its  factors,  4  and 
one  of  the  required  parts,  which  is  here  16. 

If  it  were  asked  how  many  parts,  of  16  each,  64  is  composed  of, 
it  would  be  necessary,  in  order  to  ascertain  the  number  of  these 
parts,  to  find  how  many  times  64  contains  16,  and  consequently, 
64  must  be  regarded  as  a  product,  of  which  one  of  the  factors  is 
16,  and  the  other  the  number  sought,  which  is  4. 

Whatever  then  may  be  the  object  in  view,  division  consists  in 
finding  one  of  the  factors  of  a  given  product,  when  the  other  is 
known. 

37.  The  number  to  be  divided  is  called  the  dividend,  the  fac- 
tor, that  is  known,  and  by  which  we  must  divide,  is  called  the 
divisoTf  the  factor  found  by  the  division  is  called  the  quotient, 
and  always  shows  how  times  the  divisor  is  contained  in  the 
dividend. 

It  follows  then,  from  what  has  been  said,  that  the  divisor  mul- 
tiplied hy  the  quotient,  ought  to  reproduce  the  dividend, 

38.  When  the  dividend  can  contain  the  divisor  a  great  many 
times,  it  would  be  inconvenient  in  practice  to  make  use  of  repeated 
subtraction  for  finding  the  quotient  j  it  then  becomes  necessary 
to  have  recourse  to  an  abbreviation  analogous  to  that  which  is 
given  for  multiplication.  If  the  dividend  is  not  ten  times  larger 
than  the  divisor,  which  may  be  easily  perceived  by  tlie  inspec- 
tion of  the  numbers,  and  if  the  divisor  consists  of  only  one  figure, 
the  quotient  may  be  found  by  the  table  of  Pythagoras,  since  that 
contains  all  the  products  of  factors,  that  consist  of  only  one 
figure  each.  If  it  were  asked,  for  instance,  how  many  times  8  is 
contained  in  56,  it  would  be  necessary  to  go  down  the  8th  column, 
to  the  line  in  which  56  is  found  ;  the  figure  7,  at  the  beginning 
of  this  line,  shows  the  second  factor  of  the  number  56,  or  how 
many  times  8  is  contained  in  this  number. 

We  see  by  the  same  table,  that  there  are  numbers,  which  can- 
not be  exactly  divided  by  others.  For  instance,  as  the  seventh 
line,  which  contains  all  the  multiples  of  7,  has  not  40  in  it,  it 


24  ArithmetiCi 

follows  that  40  is  not  divisible  by  7 ;  but  as  it  comes  betweea 
35  and  42,  we  see  that  the  greatest  multiple  of  7,  it  can  contain, 
is  35,  the  factors  of  which  are  5  and  7.  By  means  of  this  ele- 
mentary information,  and  the  considerations,  which  will  now  be 
offered,  any  division  whatever  may  be  performed. 

39.  Let  it  be  required,  for  example,  to  divide  1656  by  3  ;  this 
question  may  be  changed  into  another  form,  namely ;  Tojind  such 
a  numberf  that  multiplying  its  unitSf  tens,  hundreds,  ^c,  by  3,  tJie 
product  of  these  units,  tens,  hundreds,  Sfc,  may  be  the  dividend,  1656. 

It  is  plain,  that  this  number  will  not  have  units  of  a  higher 
order  than  thousands,  for,  if  it  had  tens  of  thousands,  there 
would  be  tens  of  thousands  in  the  product,  whicli  is  not  the  case. 
Neither  can  it  have  units  of  as  high  an  order  as  thousands,  for  if 
it  had  but  one  of  this  order,  the  product  would  contain  at  least  3, 
which  is  not  the  case.  It  appears  then,  that  the  thousand  in  the 
dividend  is  a  number  reserved,  when  the  hundreds  of  the  quo- 
tient were  multiplied  by  3,  the  divisor. 

This  premised,  the  figure  occupying  the  place  of  hundreds,  in 
the  required  quotient,  ought  to  be  such,  that,  when  multiplied  by 
3,  its  product  may  be  16,  or  the  greatest  multiple  of  3,  less  than 
16.  This  restriction  is  necessary,  on  account  of  the  reserved 
numbers,  which  the  other  figures  of  the  quotient  may  furnish, 
when  multiplied  by  the  divisor,  and  which  should  be  united  to 
the  product  of  the  huudreds. 

Tlie  number,  which  fulfils  this  condition  is  5 ;  but  5  hundreds, 
multiplied  by  3,  gives  15  hundreds,  and  the  dividend,  1656,  con- 
tains 16hundretls;  the  difference,  1  hundred,  must  have  come  then 
from  the  reserved  number,  arising  from  tlie  multiplication  of  the 
other  figures  of  the  quotient  by  the  divisor.  If  we  now  subtract 
the  partial  product  15  hundreds,  or  1500,  from  the  total  product 
1656,  the  remainder  156,  will  contain  the  product  of  the  units 
and  tens  of  the  quotient  by  the  divisor,  and  the  question  will  be 
reduced  to  finding  a  number,  which,  multiplied  by  3,  gives  156, 
a  question  similar  to  that,  which  presented  itself  above.  Thus 
when  the  first  figure  of  the  quotient  shall  have  been  found  in 
this  last  question,  as  it  was  in  the  first,  let  it  be  multiplied  by  the 
divisor,  then  subtracting  this  partial  product  from  the  whole 


Division,  S5 

product,  the  result  will  be  a  new  dividend,  which  may  be  treated 
in  the  same  mariner  as  the  preceding,  and  so  on,  until  the  ori- 
ginal dividend  is  exhausted. 

40.  The  operation  just  described  is  disposed  of  thus ; 


dividend  1656 
15 


3  divisor 


552  quotient 


06 
6 

0 

Tlie  dividend  and  divisor  are  separated  by  a  line,  and  another 
line  is  drawn  under  the  divisor,  to  mark  the  place  of  the  quotient. 
This  being  done,  we  take  on  the  left  of  the  dividend  the  part  16, 
capable  of  containiiig^the  divisor,  3,  and  dividing  it  by  this  num- 
ber, we  get  5  for  the  first  figure  of  the  quotient  on  the  left ',  then 
taking  the  product  of  the  divisor  by  the  number  just  found,  and 
subtracting  it  from  16,  the  partial  dividend,  we  write  under- 
neath, the  remainder,  1,  by  the  side  of  which  we  bring  down  the 
6  tens  of  the  dividend.  Considering  the  number,  as  it  now 
stands,  a  second  partial  dividend,  we  divide  it  also  by  the  divi- 
sor 3,  and  obtain  5  for  the  second  figure  of  the  quotient ;  we 
then  take  the  product  of  this  number  by  the  divisor,  and  subtract- 
ing it  from  the  partial  dividend,  get  0  for  the  remainder.  We 
then  bring  down  the  last  figure  of  the  dividend,  6,  and  divide 
this  third  partial  dividend  by  the  divisor,  3,  and  get  2  for  the  last 
figure  of  the  quotient. 

41.  It  is  manifest  that,  if  we  find  a  pai-tial  dividend,  which  can- 
not contain  the  divisor,  it  must  be  because  the  quotient  has  no 
units  of  the  order  of  that  dividend,  and  that  those  which  it  con- 
tains arise  from  the  products  of  the  divisor  by  the  units  of  t!ie 
lower  orders  in  the  quotient ;  it  is  necessary  therefore,  when- 
ever this  is  the  case,  to  put  a  0  in  the  quotient,  to  occupy  the 
place  of  the  order  of  units  that  is  wanting. 
4 


26  Jnthmetie. 

For  instance,  let  1535  be  divided  by  5. 

5 


1535 
15 


307 


035 
35 


00 


The  division  of  the  15  hundreds  of  the  dividend  by  the  divisor, 
leaving  no  remainder,  the  3  tens,  which  form  the  second  partial 
dividend,  do  not  contain  the  divisor.  Hence  it  appears,  that  tlie 
quotient  ought  to  have  no  tens  ;  consequently  this  place  must  be 
filled  with  a  cipher,  in  order  to  give  to  the  first  figure  of  the 
quotient  the  value,  it  ought  to  have,  compared  with  the  others ; 
then  bringing  down  the  last  figure  of  the  dividend,  we  form  a 
third  partial  dividend,  vvhicli,  divided  by  5,  gives  7  for  the  units 
of  the  quotient,  the  whole  of  which  is  now  307. 

42.  The  considerations,  presented  in  article  40,  apply  equally 
to  the  case,  in  which  the  divisor  consists  of  any  number  of 
figures. 

If,  for  instance,  it  were  required  to  divide  57981  by  251,  it 
would  easily  be  seen,  that  the  quotient  can  have  no  figures  of  a 
higher  order  than  hundreds,  because,  if  it  had  thousands,  tlie  div- 
idend would  contain  hundreds  of  thousands,  which  is  not  the  case  ; 
further,  the  number  of  hundreds  should  be  such,  that,  multiplied 
by  251,  the  product  would  be  579,  or  the  multiple  of  251  next 
less  than  579 ;  this  restriction  is  necessary  on  account  of  the 
reserved  numbers  which  may  have  been  furnished  by  the  multi- 
plication of  the  other  figures  of  the  quotient  by  the  divisor.  The 
number,  which  answers  to  this  condition,  is  2 ;  but  2  hundreds, 
multiplied  by  251,  give  502  hundreds,  and  the  divisor  contains 
579  ;  the  diflTerence,  77  hundreds,  arises  then  from  the  reserved 
numbers  resulting  from  the  multiplication  of  the  units  and  tens  of 
the  quotient,  by  the  divisor. 

If  we  now  subtract  the  partial  product,  502  hundreds,  or  50200, 
from  the  total  product,  57981,  the  remainder  7781,  will  contain 
the  products  of  the  units  and  tens  of  the  quotient  by  the  divisor, 


Division.  27 

and  the  operation  will  be  reduced  to  finding  a  number,  \vhich, 
multiplied  by  251,  will  give  for  a  product  7781. 

Thus,  when  the  first  figure  of  the  quotient  shall  have  been  de- 
termined, it  must  be  multiplied  by  the  divisor,  the  product  being 
subtracted  from  the  whole  dividend,  a  new  dividend  will  be  the 
result,  which  must  be  operated  upon  like  the  preceding ;  and  so 
on,  till  the  whole  dividend  is  exhausted. 

It  is  always  necessary,  for  obtaining  the  first  figure  of  the 
quotient,  to  separate,  on  the  left  of  the  dividend,  so  many  figures, 
as,  considered  as  simple  units,  will  contain  the  divisor,  and 
admit  of  this  partial  division. 

43.  Disposing  of  the  operation  as  before,  the  calculation,  just 
explained,  is  performed  in  the  following  order ; 


57981 

251 

502 

231 

778 

753 

251 

251 

000 

The  3  first  figures,  on  the  left  of  the  dividend  are  taken  to 
form  the  partial  dividend  ,•  they  are  divided  by  the  divisor,  and 
the  number  2,  thence  resulting,  is  written  in  the  quotient ;  the 
divisor  is  then  multiplied  by  this  number,  and  the  product,  502, 
is  written  under  the  partial  dividend,  579.  Subtraction  being 
performed,  the  8  tens  of  the  dividend  are  brought  down  to  the 
side  of  the  remainder,  77 ;  this  new  partial  dividend  is  then 
divided  by  the  divisor,  and  3  is  obtained  for  the  second  figure  of 
the  quotient ;  the  divisor  is  multiplied  by  this,  the  product  sub- 
tracted from  the  corresponding  partial  dividend,  and  to  the 
remainder,  25,  is  brought  down  the  last  figure  of  the  dividend,  1 ; 
this  last  partial  dividend,  251,  being  equal  to  the  divisor,  gives  1 
for  the  units  of  the  quotient. 

44.  When  the  divisor  contains  many  figures,  some  difficulty 
may  be  found  in  ascertaining  how  many  times  it  is  contained  in 


28  JnthmeUc. 

the  partial  dividends.     The  following  example  is  designed  t^ 
show  how  it  may  be  known. 


423405 

485 

3880 

873 

3540 
3395 

1455 
1455 

0000 
It  is  necessary  at  first  to  take  four  figures  on  the  left  of  the 
dividend,  to  form  a  number  which  will  contain  the  divisor;  and 
then  it  cannot  be  immediately  perceived  how  many  times  485  is 
contairied  in  4234.  To  aid  us  in  this  inquiry,  we  shall  observe, 
that  this  divisor  is  between  400  and  500  ;  and  if  it  were  exactly 
one  or  the  other  of  these  numbers,  the  question  would  be  reduced 
to  finding  how  many  times  4  hundred  or  5  hundred  is  contained 
in  the  42  hundreds  of  the  number  4234,  or,  which  amounts  to  the 
same  thing,  how  many  times  4  or  5  is  contained  in  42.  For  the 
first  of  these  numbers  we  get  10,  and  for  the  second  8,  the  quo- 
tient must  now  be  sought  between  these  two.  We  see  at  first 
that  we  cannot  employ  10,  because  this  would  imply,  that  the 
order  of  units  in  the  dividend  above  hundreds,  contained  the 
divisor,  which  is  not  the  case.  It  only  remains  then,  to  try 
which  of  the  two  numbers  9  or  8,  used  as  the  multiplier  of  485, 
gives  a  product  that  can  be  subtracted  from  4284,  and  8  is  found 
to  be  the  one.  Subtracting  from  the  partial  dividend  the  pro- 
duct of  the  divisor  multiplied  by  8,  we  get,  for  the  remainder, 
554  ;  bringing  down  then  the  6  tens  in  the  dividend,  we  form  a 
second  partial  dividend,  on  which  we  operate  as  on  the  preced- 
ing ;  and  so  with  the  others. 

45.  The  recapitulation  of  the  preceding  articles  gives  us  this 
rule.  To  divide  one  number  by  another^  place  the  divisor  on  the 
right  of  the  dividend,  separate  them  by  a  line,  and  draw  another 
line  under  the  divisor,  to  make  the  place  for  the  quotient.  Take,  on 
the  left  of  the  dividend,  as  many  figures  as  are  necessary  to  contain 


Division.  '^9 

the  divisor;  find  how  mamj  times  the  number,  expressed  by  the  first 
figure  of  the  divisor,  is  contained  in  that,  represented  by  the  first,  or 
two  first,  figures  of  the  partial  dividend  ;  multiply  this  quotient^ 
which  is  only  an  approximation,  by  the  divisor,  and,  if  the  product 
is  greater  tJian  the  partial  dividend,  take  units  from  the  quotient 
continuady,  till  it  will  give  a  product  that  can  he  subtracted  from 
the  partial  dividend ;  subtract  this  product,  and  if  the  remainder 
be  greater  than  the  dividend,  it  will  be  a  proof  that  the  quotient  has 
been  too  much  diminished  ;  and,  consequently,  it  must  be  increased. 
By  the  side  of  the  remainder  bring  down  the  next  figure  of  the 
dividend,  and  find,  as  before,  how  many  times  this  partial  dividend 
contains  the  divisor  ;  continue  thus,  until  all  the  figures  of  the  given 
dividend  are  brought  down.  When  a  partial  dividend  occurs,  which 
does  not  contain  the  divisor,  it  is  necessary,  before  bringing  down 
another  figure  of  the  dividend,  to  put  a  cipher  in  the  quotient. 

46.  The  operations  required  in  division  may  be  made  to  oc- 
cupy a  less  space,  by  performing  mentally  the  subtraction  of  the 
products  given  by  the  divisor  and  each  figure  of  the  quotient,  as 
is  exhibited  in  the  following  example ; 
1755     j     39 

195     I     45 

000 

After  having  found  that  the  first  partial  dividend  contains  4 
times  the  divisor,  39,  we  multiply  at  first  the  9  units  by  4,  which 
gives  36  ;  and,  in  order  to  subtract  this  product  from  the  partial 
dividend,  we  add  to  the  5  units  in  the  dividend,  4  tens,  making 
their  sum  45,  from  which  taking  36,  9  remain.  We  then  re- 
serve 4  tens  to  join  them,  in  the  mind,  to  12,  the  product  of  the 
quotient  by  the  tens  in  the  divisor,  making  the  sum  16  ;  in  taking 
this  sum  from  17,  we  take  away  the  4  tens,  with  which  we  had 
augmented  the  units  of  the  dividend,  in  order  to  perform  the 
preceding  subtraction.  We  then  operate  in  the  same  manner  on 
the  second  partial  dividend,  195,  saying;  9  times  5  make  45, 
taken  from  45,  nought  remains,  then  5  times  3  make  15,  and  4 
tens,  reserved,  make  19,  taken  from  19,  nought  remains. 

We  see  sufficiently  by  this  in  what  manner  we  are  to  per- 
form any  other  example,  however  complicated. 


30 


Arithmetic. 


47.  Division  is  also  abbreviated  wben  the  dividend  and  divi- 
sor are  terminated  by  ciphers,  because  we  can  strike  out,  from 
the  end  of  each,  as  many  ciphers  as  are  contained  in  the  one  that 
has  the  least  number. 

If,  for  instance,  84000  were  to  be  divided  by  400,  these  num- 
bers may  be  reduced  to  840  and  4,  and  tbe  quotient  would  not  be 
altered  j  for  we  should  only  have  to  change  the  name  of  the 
units,  since,  instead  of  84000,  or  840  hundreds,  and  400,  or  4 
hundreds,  we  should  have  840  units  and  4  units,  and  the  quotient 
of  the  numbers  840  and  4  is  always  tlie  same,  whatever  may  be 
the  denomination  of  their  units. 

It  may  also  be  remarked  that,  in  striking  out  two  ciphers  at 
the  end  of  the  given  numbers,  tliey  have  been,  at  the  same  time, 
both  of  them  divided  by  100 ;  for  it  follows  from  article  31,  that 
in  striking  out  1,  2  or  3  ciphers  on  the  right  of  any  number,  the 
number  is  divided  by  10,  or  100,  or  1000,  &c. 


Examplt 

s  in  Division. 

144 

3               16512 
48                 2752 

344 

3049164 
53956 

6274 

24 

48 

486 

GO 

0000 

37644 

00000 

Divide  49561776        by  5137 

yjyjyjyjyj 

Ans.  96 

48. 

27489000         by  350. 

Alls,  7854. 

2150596750    by  125. 

Ans.  17204774. 

32500000         by  520. 

Ans.  62500. 

10392304800  by  20 

. 

Ans.  9.t 

980762. 

48.  Division  and  multiplication  mutually  prove  each  other, 
like  subtraction  and  addition,  for  according  to  the  definition  of 
division,  (36),  we  ougbt,  by  dividing  the  product  by  one  of  the 
factors,  to  find  the  other ;  and  multiplying  the  divisor  by  the 
quotient  we  ought  to  reproduce  the  dividend  (37). 


Fractions. 
49.  Division  cannot  always  be  exactly  performed,  because 
any  number  whatever  of  units  taken  a  certain  number  of  times, 
does  not  always  compose  any  other  number  whatever.    Exam- 


Division.  81 

pies  of  this  have  already  been  seen  in  tlie  table  of  Pythagoras, 
vvhicli  contains  only  the  product  of  the  9  first  numbers,  multiplied 
two  and  two,  but  does  not  contain  all  the  numbers  between  1 
and  81,  the  first  and  last  numbers  in  it.  The  metiiod  hitherto 
given  shows  then,  only  how  to  find  the  greatest  multiple  of  the 
divisor,  that  can  be  contained  in  the  dividend. 

If  we  divide  239  by  8,  according  to  the  rule  in  article  46. 
239  8 

79  29 

7 
we  have,  for  the  last  partial  dividend,  the  number  79,  which  does 
not  contain  8  exactly,  but  which,  falling  between  the  two  numbers, 
72  and  80,  one  of  which  contains  the  divisor,  8,  nine  times,  and 
the  other  ten,  shows  us  that  the  last  part  of  the  quotient  is  greater 
than^9,  and  less  than  10,  and  consequently,  that  the  whole  quo- 
tient is  between  29  and  30.  If  we  multiply  the  unit  figure  of 
the  quotient,  9,  by  the  divisor,  8,  and  subtract  the  product  from 
the  last  partial  dividend,  79,  the  remainder,  7,  will  evidently  be 
the  excess  of  the  dividend,  239,  above  the  product  of  the  factors, 
29  and  8.  Indeed,  having,  by  the  different  parts  of  the  operation, 
subtracted  successively  from  the  dividend,  239,  the  product  of 
each  figure  of  the  quotient  by  the  divisor,  we  have  evidently  sub- 
tracted the  product  of  the  whole  quotient  by  the  divisor,  or  232  ; 
and  the  remainder,  7,  less  than  the  divisor,  proves,  that  232  is 
the  greatest  multiple  of  8,  that  can  be  contained  in  239. 

50.  It  must  be  perceived,  after  wliat  has  been  said,  that  to 
reproduce  any  dividend,  we  must  add  to  the  product  of  the  divi- 
sor by  the  quotient,  the  sum  which  remains  when  the  divisor 
cannot  be  performed  exactly. 

51.  If  we  wished  to  divide  into  eight  equal  parts  a  sum  of 
whatever  nature,  consisting  of  239  units,  we  could  not  do  it  with- 
out using  parts  of  units  or  fractions.  Thus,  when  we  liave  taken 
from  the  number  239,  the  8  times  29  ujiits  contained  in  it,  there 
will  remain  7  units,  to  be  divided  into  8  parts;  to  do  this,  we 
may  divide  each  of  these  units,  one  after  the  other,  into  8  parts, 
and  then  take  one  part  out  of  each  unit,  wliich  will  give  7  parts 
to  be  joined  to  the  29  whole  units,  to  form  the  eighth  part  of 
239,  or  the  exact  quotient  of  this  number,  by  8. 


S2  Jirithmelic, 

The  same  reasoning  may  be  applied  to  every  other  example 
of  division  in  which  there  is  a  remainder,  and  in  this  case  the 
quotient  is  composed  of  two  parts ;  one,  consisting  of  whole 
units,  while  the  other  cannot  be  obtained,  until  the  concrete  or 
material  units  of  the  remainder  have  been  actually  divided  into 
the  number  of  parts  denoted  by  the  divisor  ;  without  this  it  can 
only  be  indicated  by  supposing,  a  imit  of  the  dividend  to  be  divid- 
ed into  as  many  parts  as  there  are  units  in  the  divisor,  and  so  many 
of  these  parts,  as  there  are  units  in  the  remainder,  taken  to  complete 
the  quotient  required. 

52.  In  general,  when  we  have  occasion  to  consider  quantities 
less  than  unity,  we  suppose  unity  divided  into  a  certain  number 
of  parts,  sufficiently  small  to  be  contained  a  certain  number  of 
times  in  these  quantities,  or  to  measure  them.  In  the  idea  thus 
formed  of  tlieir  magnitude  there  are  two  elements,  namely,  the 
number  of  times  the  measuring  part  is  contained  in  unity,  and 
the  number  of  these  parts  found  in  the  quantities. 

A  nomenclature  has  been  made  for  fractions,  which  answers 
to  this  manner  of  conceiving  and  representing  them. 

That  whicli  results  from  the  division  of  unity 
into  2  parts  is  called  a  moiety  or  half, 
into  3  parts  a  third, 

into  4  parts  a  quarter  or  fourth, 

into  5  parts  affth, 

into  6  parts  a  sixth, 

and  so  on,  adding  after  the  two  first,  the  termination  th  to  the  num- 
ber, which  denotes  how  many  parts  are  supposed  to  be  in  unity. 

Every  fraction  then  is  expressed  by  two  numbers  ;  the  first, 
which  shows  how  many  parts  it  is  composed  of,  is  called  the 
numerator,  and  the  other  which  shows  how  many  of  these  parts 
are  necessary  to  form  an  unit,  is  called  the  denominator,  because 
the  denomination  of  the  fraction  is  deduced  from  it.  Five  sixths 
of  an  unit  is  a  fraction,  the  numerator  of  which  is  five,  and  the 
denominator  six. 

Tlie  numerator  and  the  denominator  together  are  called  the  two 
terms  of  the  fraction. 

Figures  are  used  to  shorten  the  expression  of  fractions,  the 


Fractians.  33 

denominator  being  written  under  the  numerator,  and  separated 
from  it  by  a  line, 

one  third  is  written  -I, 
Jive  sixths  f. 

53.  According  to  the  meaning  attached  to  the  words,  numeral 
tor  and  denominatorf  it  is  plain,  that  a  fraction  is  increasedf  hy 
increasing  its  mimerator,  without  changing  its  denominator ;  for 
this  last,  as  it  shows  into  how  many  parts  unity  is  divided,  deter- 
mines the  magnitude  of  these  parts,  which  continues  the  same, 
while  the  denominator  remains  unchanged  ;  and  by  augmenting 
the  numerator  the  number  of  these  parts  is  augmented,  and  con- 
sequently the  fraction  increased.  It  is  thus,  for  instance,  that  ^ 
exceeds  |^,  and  that  ||  exceeds  W* 

It  follows  evidently  from  this,  tliat  hy  repeating  the  numerator 
2,  3,  or  any  number  of  times,  without  altering  the  denominator, 
we  repeat,  a  like  number  of  times,  the  quantity  expressed  hy  the 
fraction,  or  in  other  words  midtiply  it  by  this  number ;  for  we 
make  2,  3,  or  any  number  of  times,  as  many  parts,  as  it  had 
before,  and  these  parts  have  remained  each  of  the  same  value. 

The  fraction  |,  then,  is  the  triple  of  |,  and  ^  the  double  of  /y . 

A  fraction  is  diminished  hy  diminishing  its  numerator,  without 
changing  its  denominator,  since  it  is  made  to  consist  of  a  less 
number  of  parts  than  it  contained  before,  and  these  parts  retain 
the  sams  value.  Whence,  if  the  numerator  be  divided  by  2,  3,  or 
any  number,  without  the  denominator  being  altered,  the  fraction  is 
made  a  like  number  of  times  smaller,  or  is  divided  by  that  number, 
for  it  is  made  to  contain  2,  3,  or  any  number  of  times  less 
parts  than  it  contained  before,  and  these  parts  remain  of  the 
same  value.    Thus  |  is  a  third  of  |,  and  ^\  is  half  of  ^4. 

54.  On  the  contrary,  a  fraction  is  diminished,  when  its  de- 
nominator is  increased  without  changing  its  numerator  j  for 
then  more  parts  are  supposed  in  an  unit,  and  consequently  they 
must  be  smaller,  but,  as  only  the  same  number  of  them  are  taken 
to  form  the  fraction,  the  amount  in  this  case  must  be  a  less  quan- 
tity than  in  the  first.     Thus  |  is  less  than  |,  and  j%  than  ±. 

Hence  it  follows,  that  if  the  denominator  of  a  fraction  he  multi- 
plied hy  2,  3,  or  any  number,  without  the  numerator  being  changed, 


34  Arithmetic. 

the  fraction  becomes  a  like  number  of  times  smaller  ^  or  is  divided  by 
that  number,  for*  it  is  composed  of  the  same  number  of  parts  as 
before,  but  each  of  them  has  become  2,  3,  or  a  certain  number 
of  times  less.     The  fraction  |  is  half  of  |,  and  -j^  the  third  of  |. 

*i  fraction  is  increased  when  its  denominator  is  diminished  with- 
out the  numerator  being  changed  ;  because,  as  unity  is  supposed  to 
be  divided  into  fewer  parts,  each  one  becomes  greater,  and  their 
amount  is  therefore  greater. 

Whence,  if  the  denominator  of  a  fraction  be  divided  by  2,  3,  or 
any  other  number^  the  fraction  will  be  made  a  like  number  of  times 
greater,  or  will  be  multiplied  by  that  number  ;  for  the  number  of 
parts  remains  the  same,  and  each  one  becomes  2,  3,  or  a  certain 
number  of  times  greater  than  it  was  before*  According  to  this 
I  is  triple  of  ^^j  and  |  the  quadruple  of  ^*^. 

It  may  be  remarked,  that  to  suppress  the  denominator  of  a 
fraction  is  the  same  as  to  multiply  the  fraction  by  that  number. 
For  instance,  to  supress  the  denominator  3  in  the  fraction  -|  is  to 
change  it  into  2  wliole  ones,  or  to  multiply  it  by  3. 

55.  The  preceding  propositions  may  be  recapitulated  as  follows  ; 

By  dividing"^}  the  numerator,  the  fraction  is  { [H^fj^^J;^^- 

By  Tvidhig"^}*^^  denominator,  the  fraction  is { ^|;;;fipi;,d. 

56.  The  first  consequence  to  be  drawn  from  this  table  is,  that 
the  operations  performed  on  the  denominator  produce  effects  of 
an  inverse  or  contrary  nature  with  respect  to  the  value  of  the 
fraction.  Hence  it  results,  that,  if  both  the  numerator  and  denom- 
inator of  a  fraction  be  midtiplied  at  the  same  time,  by  the  same 
number,  the  value  of  the  fraction  will  not  be  altered  ;  for  if,  on  the 
one  hand,  multiplying  the  numerator  makes  the  fraction  2,  3,  &c. 
times  greater,  so  on  the  other,  by  the  second  operation,  the  half 
or  third  part,  &c.  of  it  is  taken  ;  in  otiicr  words  it  is  divided  by 
the  same  number,  by  which  it  had  at  first  been  multiplied. 
Thus  -J-  is  equal  to  -j?^,  and  /^  is  equal  to  1|. 

57.  It  is  also  manifest  that,  if  both  the  numerator  and  denomi- 
nator of  a  fraction  be  divided,  at  the  same  time,  by  the  same  num- 
ber, tJie  value  of  the  fraction  will  not  be  altered  ;  for  if,  on  the  one 
hand,  hy  dividing  the  numerator  the  fraction  is  made  2,  3,  &c» 


Fractions.  35 

times  smaller  ;  on  the  other,  by  the  second  operation,  the  double, 
triple,  &c.  is  taken ;  in  short  it  is  multiplied  by  the  same  num- 
ber, by  which  it  was  at  first  divided.  Thus  the  fraction  |  is  equal 
to  i,  and  I  is  equal  to  ^. 

58.  It  is  not  with  fractions  as  with  whole  numbers,  in  which  a 
magnitude,  so  long  as  it  is  considered  with  relation  to  the  same 
unit,  is  susceptible  of  but  one  expression.  In  fractions  on  the 
contrary,  the  same  magnitude  can  be  expressed  in  an  infinite 
number  of  ways.     For  instance,  the  fractions, 

12        3       4         5  6  7    ,     Afp 

2*    4»    ^'   ¥'    TTT'    TT'    TT*    "^^* 

in  each  of  which  the  denominator  is  twice  as  great  as  the  nume- 
rator, express,  under  different  forms,  the  half  of  an  unit.  The 
fractions,  -^-,  f,  |,  VV'  tt'  tV»  2^'  ^^^ 

of  which  the  denominator  is  three  times  as  great  as  the  numera- 
tor, represent  each  the  third  part  of  an  unit.  Among  all  the 
forms,  which  the  given  fraction  assumes,  in  each  instance,  the 
first  is  the  most  remarkable,  as  bcirjg  the  most  simple  ;  and,  con- 
sequently, it  is  well  to  know  how  to  find  it  from  any  of  the 
others.  It  is  obtained  by  dividing  the  two  terms  of  the  others 
by  the  same  number,  which,  as  has  already  been  shown,  docs  not 
alter  their  value.  Thus  if  we  divide  by  7  the  two  terms  of  the 
fraction  J'^,  we  come  back  to  | ;  and,  perfoj'ming  the  same  oper- 
ation on  ^\,  we  get  i. 

59.  It  is  by  following  this  process,  that  a  fraction  is  reduced 
to  its  most  simple  terms  ;  it  cannot,  however,  be  applied  except  to 
fractions,  of  which  the  numerator  and  denominator  are  divisible 
by  the  same  number  ;  in  all  other  cases  the  given  fraction  is  the 
most  simple  of  all  those,  that  can  represent  the  quantity  it  ex- 
presses. Thus  the  fractions  \,  y\,  if,  the  terms  of  which  can- 
not be  divided  by  the  same  number,  or  have  no  common  divisor, 
are  irreducible,  and,  consequently,  cannot  express,  in  a  more  sim- 
ple manner,  the  magnitudes  which  they  represent. 

60.  Hence  it  follows,  that  to  simplify  a  fraction,  we  must 
endeavour  to  divide  its  two  terms  by  some  one  of  the  numbers, 
2,  3,  &c  ;  but  by  this  uncertain  mode  of  proceeding  it  will  not 
be  always  possible  to  come  at  the  most  simple  terms  of  the  given 
fraction,  or  at  least,  it  will  often  be  necessary  to  perform  a  great 
number  of  operations. 


36  Jnthmetic, 

If,  for  instance,  tlie  fraction  ||  were  given,  it  may  be  seen  at 
once,  that  each  of  its  terms  is  a  multiple  of  2,  and  dividing  them 
by  this  number,  we  obtain  ^| ;  dividing  these  last  also  by  2,  we 
obtain  -^\.  Although  much  more  simple  now  than  at  first,  this 
fraction  is  still  susceptible  of  reduction,  for  its  two  terms  can  be 
divided  by  3,  and  it  then  becomes  |. 

If  we  observe,  that  to  divide  a  number  by  2,  then  the  quotient 
by  2,  and  then  the  second  quotient  by  3,  is  the  same  thing  as  to 
divide  the  original  number  by  the  product  of  the  numbers,  2, 2, 
and  3,  which  amounts  to  12,  we  shall  see  that  the  three  above 
operations  can  be  performed  at  once  by  dividing  the  two  terms 
of  the  given  fraction  by  12,  and  we  shall  again  have  f . 

The  numbers  2,  3,  4,  and  12,  each  dividing  the  two  numbers 
24  and  84  at  the  same  time,  are  the  common  divisors  of  these 
numbei*s ;  but  12  is  the  most  worthy  of  attention,  because  it  is 
the  greatest,  and  it  is  by  employing  tlie  greatest  common  divism^ 
of  the  two  terms  of  the  given  fraction,  that  it  is  reduced  at  once 
to  its  most  simple  terms.  We  have  then  this  important  prob- 
lem to  solve,  two  numbers  being  given,  to  find  their  greatest  com- 
mon divisorj;. 

61.  We  arrive  at  the  knowledge  of  the  common  divisor  of  two 
numbers  by  a  sort  of  trial  easily  made,  and  which  has  this  re- 
commendation, that  each  step  brings  us  nearer  and  nearer  to 
the  number  sought.  To  explain  it  clearly,  I  will  take  an  example. 

Let  the  two  numbers  be  637  and  143.  It  is  plain,  that  the 
greatest  common  divisor  of  these  two  numbere  cannot  exceed  the 
smallest  of  them ;  it  is  proper  then  to  try  if  the  number  143, 
which  divides  itself  and  gives  1  for  the  quotient,  will  also  divide 
the  number  637,  in  which  case  it  will  be  the  greatest  common 
divisor  sought.  In  the  given  example  this  is  not  the  case ;  we 
obtain  a  quotient  4,  and  a  remainder  65. 

Now  it  is  plain,  that  every  common  divisor  of  the  two  num- 
bers, 143  and  637,  ought  also  to  divide  65,  tlie  remainder  result- 
ing from  their  division  ;  for  the  greater,  637,  is  equal  to  the 


t  What  is  here  called  the  greatest  common  divisor,  is  sometimes 
called  the  o;reatest  common  measure^ 


Fractims,  S7 

less,  143,  multiplied  by  4,  plus  the  remainder,  65,  (50)  ;  now  in 
dividing  637  by  the  common  divisor  sought,  we  shall  have  an 
exact  quotient ;  it  follows  then,  that  we  must  obtain  a  like  quo- 
tient, by  dividing  the  assemblage  of  parts,  of  which  637  is  com- 
posed, by  the  same  divisor ;  but  the  product  of  143  by  4  must 
necessarily  be  divisible  by  the  common  divisor,  which  is  a  factor 
of  143,  and  consequently  the  other  part,  65,  must  also  be  divisi- 
ble by  the  same  divisor ;  otherwise  the  quotient  would  be  a  whole 
number  accompanied  by  a  fraction,  and  consequently  could  not 
be  equal  to  the  whole  number,  resulting  from  the  division  of 
637  by  the  common  divisor.  By  the  same  reasoning,  it  may  be 
proved  in  general,  that  every  common  divisor  of  two  numbers  must 
also  divide  the  remainder  resulting  from  the  divisio7i  of  the  greater 
of  the  two  by  the  less. 

According  to  this  principle,  we  see,  that  the  common  divisor 
of  the  numbers,  637  and  143,  must  also  be  the  common  divisor 
of  the  numbers  143  and  65 ;  but  as  the  last  cannot  be  divided  by 
a  number  greater  than  itself,  it  is  necessary  to  try  65  first. 
Dividing  143  by  65,  we  find  a  quotient  2,  and  a  remainder  13  j 
65  then  is  not  the  divisor  sought.  By  a  course  of  reasoning, 
similar  to  that  pursued  with  regard  to  the  numbers,  637,  143, 
and  the  remainder,  resulting  from  their  division,  65,  it  will  be  seen 
that  every  common  divisor  of  143  and  65  must  also  divide 
the  numbers  65  and  13;  now  the  greatest  common  divisor  of 
these  two  last  cannot  exceed  13,  we  must  therefore  try,  if  13  will 
divide  65,  which  is  the  case,  and  the  quotient  is  5  ;  then  13  is 
the  greatest  common  divisor  sought. 

We  can  make  ourselves  certain  of  its  possessing  this  property 
by  resuming  the  operations  in  an  inverse  order,  as  follows  j 

As  13  divides  65  and  13,  it  will  divide  143,  which  consists  of 
twice  65  added  to  13;  as  it  divides  65  and  143,  it  will  divide 
637,  which  consists  of  4  times  143  added  to  65  ;  13  then  is  the 
common  divisor  of  the  two  given  numbers.  It  is  r^lso  evident, 
by  the  very  mode  of  finding  it,  that  there  can  be  no  common 
divisor  greater  than  IS,  since  13  must  be  divided  by  it. 

It  is  convenient  in  practice,  to  place  the  successive  divisions 
one  after  the  other,  and  to  dispose  of  the  operation  as  may  be  seen 
in  the  following  example ; 


^8 


^Arithmetic. 


637 

572 
"65 


143 
4(130 

_65 

2|65 

13 

0 

115 


SM. 


the  quotients,  4,  2,  5,  being  separated  from  the  other  figures. 

The  reasoning  employed  in  the  preceding  example,  may  be 
applied  to  any  numbers,  and  thus  conduct  us  to  this  general  rule. 
The  greatest  common  dimsor  of  two  nnmbers  irill  be  found,  by 
dividing  the  greater  by  the  less  ;  then  the  less  by  the  remainder  of 
the  first  division;  then  this  remainder,  by  the  remainder  of  the 
second  division;  then  this  serond  remainder  by  the  third,  m'  that  of 
the  third  division ;  and  so  on,  till  we  arrive  at  an  exact  quotient ; 
the  last  divisor  will  be  the  common  divisor  sotight. 

62.  See  two  examples  of  the  operation. 

752 


9024 

7520 

3760 
2|S008 

1504 
2J1504 

.1 

1504 

752 

00 

752  then  is  the  greatest  common  divisor  of  9024  and  3760. 


47 

44 

3 

2 

T9I44 

1  13 

14  j2 

1   I2 

3 

14 
12 

2 

1 

0 

937 

47 

467 

423 
"44 


By  this  last  operation  we  see  that  the  greatest  common  divi- 
sor of  937  and  47,  is  1  only,  that  is,  these  two  numbers  pro- 
perly speaking  have  no  common  divisor,  since  all  whole  num- 
bers, like  them,  are  divisible  by  1. 

We  may  easily  satisfy  ourselves,  that  the  rule  of  the  preceding 
article  must  necessarily  lead  to  this  result,  whenever  the  given 
numbers  have  no  common  divisor ;  for  the  remainders,  each 
being  less  than  the  corresponding  divisor,  become  less  and  less 
every  operation,  and  it  is  plain,  that  the  division  will  continue 
as  long  as  there  is  a  divisor  greater  than  unity. 

63.  After  these  calculations,  the  fractions  ^  and  ^^  s  0^  p^n 
be  at  once  reduced  to  their  most  simple  terms,  by  dividing  the 
terms  of  the  first  by  their  common  divisor,  13,  and  the  terms  of 
the  second,  bv  their  common  divisor,  752 ;  we  thus  obtain  ^-J- 


fradions.  39 

and  Jj.  As  to  the  fraction,  //y,  it  is  altogether  irreducible, 
since  its  terms  have  no  common  divisor  but  unity. 

64.  It  is  not  always  necessary  to  find  the  greatest  common 
divisor  of  the  given  fraction ;  there  are,  as  has  before  been 
remarked,  reductions,  which  present  themselves  without  this 
preparatory  step. 

Every  number  terminated  by  one  of  the  figures  0,  2,  4,  6,  8, 
is  necessarily  divisible  by  2  ;  for  in  dividing  any  number  by  2, 
only  1  can  remain  from  the  tens  j  the  last  partial  division  can 
be  performed  on  the  numbers  0,  2, 4,  6,  8,  if  the  tens  leave  no 
remainder,  and  on  the  numbers  10,  12, 14,  16,  18,  if  they  do, 
and  all  these  numbers  are  divisible  by  2. 

The  numbers  divisible  by  2,  are  called  even  numbers,  because 
they  can  be  divided  into  two  equal  parts. 

Also,  every  number  terminated  on  the  vight  by  a  cipher,  or 
by  5,  is  divisible  by  5,  for  when  the  division  of  the  tens  by  5  lias 
been  performed,  the  remainder,  if  there  be  one,  must  necessarily 
be  either  1,  2,  3,  or  4,  the  remaining  part  of  the  operation  will 
be  performed  on  the  numbers  0,  5,  10,  15,  20,  25,  30,  35,  40,  or 
45,  all  of  which  are  divisible  by  5. 

The  numbers,  10,  100,  1000,  &c.  expressed  by  unity  followed 
by  a  number  of  ciphers,  can  be  resolved  into  9  added  to  1,  99 
added  to  1,  999  added  to  1,  and  soon;  and  the  numbers,  9,  99, 
999,  &c.  being  divisible  by  3,  and  by  9,  it  follows  that,  if  num- 
bers of  the  form  10,  100,  1000,  &c.  be  divided  by  3  or  9,  the 
remainder  of  the  division  will  be  1. 

Now  every  number,  which,  like  20,  300  or  5000,  is  expressed 
by  a  single  significant  figure  followed  on  the  right,  by  a  number 
of  ciphei's,  can  be  resolved  into  several  numbers  expressed  by 
unity,  followed  on  the  riglit  by  a  number  of  ciphers  ;  20  is  equal 
to  10  added  to  lO;  300,  to  100  added  to  100  added  to  100;  5000,  to 
1000  added  to  1000  added  to  1000  added  to  1000  added  to  1000  ; 
and  so  with  others.  Eence  it  follows,  that  if  20,  or  1 0  added  to  10, 
be  divided  by  3  or  9,  the  remainder  will  be  1  added  to  1,  or  2; 
if  300,  or  100  added  to  100  added  to  100,  be  divided  by  3  or  9, 
the  remainder  will  be  1  added  to  1  added  to  1,  or  3. 

fn  general,  if  we  resolve  in  the  same  manner  a  number  ex- 


40  Jtrithnutic, 

pressed  by  one  significant  figure,  followed,  on  the  right,  by  a  num- 
ber of  ciphers,  in  order  to  divide  it  by  3  or  9  j  the  remainder  of 
this  division  will  be  equal  to  as  many  times  1,  as  there  are  units 
in  the  significant  figure,  that  is,  it  will  be  equal  to  the  significant 
figure  itself.  Now  any  number  being  resolved  into  units,  tens, 
hundreds,  &c.  is  formed  by  the  union  of  several  numbers  ex- 
pressed by  a  single  significant  figure  ;  and,  if  each  of  these  last  be 
divided  by  3  or  9,  the  remainder  will  be  equal  to  one  of  the  sig- 
nificant figures  of  the  given  number  ;  for  instance,  the  division 
of  hundreds  will  give,  for  a  remainder,  the  figure  occupying  the 
place  of  hundreds  ;  that  of  tens,  the  figure  occupying  the  place 
of  tens  ;  and  so  of  the  others.  If  then,  the  sum  of  all  these 
remainders  be  divisible  by  3  or  9,  the  division  of  the  given  num- 
ber by  3  or  9  can  be  performed  exactly  ;  whence  it  folio w^s,  that 
if  the  sum  of  the  figures,  constituting  any  number,  be  divisible 
by  3  or  9,  the  number  itself  is  divisible  by  3  or  9. 

Thus  the  numbers,  423,  4251,  15342,  are  divisible  by  3,  be- 
cause the  sum  of  the  significant  figures  is  9  in  the  first,  12  in  the 
second,  and  15  in  the  third. 

Also,  621,  8280,  934218,  are  divisible  by  9,  because  the  sum  of 
the  significant  figures  is  9  in  the  first,  18  in  the  second,  and  27  in 
the  third. 

It  must  be  observed,  that  every  number  divisible  by  9  is  also 
divisible  by  3,  although  every  number  divisible  by  3,  is  not  also 
divisible  by  9. 

Observations  might  be  made  on  several  other  numbers  analo- 
gous to  those  just  given  on  2,  3,  5  and  9 ;  but  this  would  lead  me 
too  far  from  the  subject. 

The  numbers  1,  3,  5,  7,  11,  13,  17,  &c.  which  can  be  divided 
only  by  themselves,  and  by  unity,  are  c&]\ei\ prime  numbers;  two 
numbers,  as  12  and  35,  having,  each  of  them,  divisors,  but 
neither  of  tliem  any  one,  that  is  common  to  it  with  the  otlier,  are 
called  prime  to  each  other. 

Consequently,  the  numerator  and  denorainjitor  of  an  irreduci- 
ble fraction  are  prime  to  eagh  other. 

Examples  for  practice  tinder  Jlrticle  Gl. 
What  is  the  greatest  common  divisor  of  24  and  36  ?   <.9ns.  12. 


Fractions,  41 

What  is  the  greatest  common  divisor  of  35  and  100  ?    Ans,  5. 
What  is  the  greatest  common  divisor  of  312  and  504  ? 

Jins.  24. 

Examples  for  practice  under  articles  57,  SB  and  60. 

Reduce  f  |  to  its  most  simple  terms.  Ans.  ^» 

Reduce  ^VA  to  its  most  simple  terms.  Ans.  |. 

Reduce  ^y^  to  its  most  simple  terms.  Ans.  ^. 

Reduce  4?^  to  its  most  simple  terms.  Ans.  f^^.. 

Reduce  |i|  to  its  most  simple  terms.  Ans,  |.. 

Reduce  |||§^  to  its  most  simple  terms.  Ans.  ^1. 

65.  After  tliis  digression  we  will  resume  the  examination  of 
the  table  in  article  55. 

By  multiplying  |  the  numerator,  the  fraction  is  1"?"%^, 
By  dividing        J  (_  divided. 

By  multiplying   |  ^^^  denominator,  the  fraction  is  (  ^^^^.^r'  a 
By  dividing        J  (_  multiplied, 

that  we  may  deduce  trom  it  some  new  inferences. 

We  see  at  once,  by  an  inspection  of  this  table,  that  a  fraction 
can  be  multiplied  in  two  ways,  namely,  by  multiplying  its  nu- 
merator, or  dividing  its  denominator,  and  that,  it  can  also  be 
divided  in  two  ways,  namely,  by  dividing  its  numerator,  or  mul- 
tiplying its  denominator ;  hence  it  follows,  that  multiplication 
alone,  according  as  it  is  performed  on  the  numerator  or  denomi- 
nator, is  sufficient  for  the  multiplication  and  division  of  fractions 
by  whole  numbers.  Thus  j%f  multiplied  by  7  units,  makes  fj ; 
|,  divided  by  3,  makes  /y* 

Examples  for  practice. 

Multiply  I  by  5.        Ans.  \°..  Divide  |  by  3.       Ms.  |. 

Multiply  /-r  by  4.       Ans.  if.  Divide  ^\.  by  6.     Ans.  ^\. 

Multiply  j\  by  6.       Ans.  |.  Divide  |  by  10.     Ans.  ^\, 

Multiply  I-  by  30.       Ans.  ^f  <>.  Divide  |  by  8.       Ans.  y\. 

Multiply  ^V  Ijy  5.       Ans.  \.  Divide  f"  by  4.     Ans.  \. 

Multiply  ^^  by  9.       Ans.  |.  Divide  f |  by  4.     Ans.  i. 

66.  The  doctrine  of  fractions  enables  us  to  generalize  the 
difinition  of  multiplication  given  in  article  21.    When  the  multi- 

6 


42  drithmetic 

plier  is  a  whole  number,  it  shows  how  raany  times  the  multipli- 
cand is  to  be  repeated  ;  but  the  term  multiplication,  extended  to 
fractional  expressions,  does  not  always  imply  augmentation,  as 
in  the  case  of  whole  numbers.  To  comprehend  in  one  state- 
ment every  possible  case,  it  may  be  said,  that  to  multiply  one 
numi)er  by  another  is,  to  form  a  number  by  means  of  thefrst,  in  the 
same  manner  as  the  second  is  formed,  by  means  of  unity.  In  real- 
ity, when  it  is  required  to  multiply  by  2,  by  3,  &c.  the  product 
consists  of  twice,  three  times,  &c.  the  multiplicand,  in  the  same 
way  as  the  multiplier  consists  of  two,  three,  &c.  units  ;  and  to 
multiply  any  number  by  a  fraction  |  for  example,  is  to  take  the 
fifth  part  of  it,  because  the  multiplier  ^,  being  tbe  fifth  part  of 
unity,  shows  that  the  product  ought  to  be  the  fifth  part  of  the 
multiplicand*. 

Also,  to  multiply  any  number  by  4  is  to  take  out  of  this  num- 
ber or  the  multiplicand,  a  part,  which  shall  be  four  fifths  of  it,  or 
equal  to  four  times  one  fifth. 

Hence  it  follows,  that  the  object  in  multiplying  by  a  fraction, 
whatever  may  be  the  multiplicand^  is,  to  take  out  of  the  multiplicand 
apart,  denoted  by  the  multiplying  fraction ;  and  that  this  opera- 
tion is  composed  of  two  others,  namely,  a  di^ision  and  a  multi- 
plication, in  which  the  divisor  and  multiplier  are  whole  numbers. 

Thus,  for  instance,  to  take  ^  of  any  number,  it  is  first  neces- 
sary to  find  the  fifth  part,  by  dividing  the  number  by  5,  and  to 
repeat  this  fifth  part  four  times,  by  multiplying  it  by  4. 

We  see,  in  general,  that  the  multiplicand  must  be  divided  by  the 
denominator  of  the  midtiplijing  fraction,  and  the  quotient  he  multi- 
plied by  its  numerator. 

The  multiplier  being  less  than  unity,  the  product  will  be  small- 
er than  the  multiplicand,  to  which  it  would  be  only  equal,  if  the 
multiplier  were  1. 

67.  If  the  multiplicand  be  a  whole  number  divisible  by  5,  for 

*  We  are  led  to  this  statement,  by  a  question  which  often  presents 
itself;  namely,  where  the  price  of  any  quantity  of  a  thing  is  required, 
the  price  of  the  unity  of  the  thing  being  known.  The  question  evi- 
dently remains  the  same,  whether  the  given  quantity  be  greater  or 
less  than  this  unity. 


Fractions,  43 

instance,  35,  the  fifth  part  will  be  7  ,*  this  result,  multiplied  by  4, 
will  give  28  for  the  ^  of  35,  or  for  the  product  of  35  by  |.  If 
the  multiplicand,  always  a  whole  number,  be  not  exactly  divisi- 
ble by  5,  as,  for  instance,  if  it  were  32,  the  division  by  5  will 
give  for  a  quotient  6|;  this  quotient  repeated  4  times  will  give24|. 

This  result  presents  a  fraction  in  which  the  numerator  exceeds 
the  denominator,  but  this  may  be  easily  explained.  The  ex- 
pression |,  in  reality  denoting  8  parts,  of  which  5,  taken 
together,  make  unity,  it  follows,  that  |  is  equivalent  to  unity 
added  to  three  fifths  of  unity,  or  1| ;  adding  this  part  to  the  24 
units,  we  have  25|  for  the  value  of  -J  of  32. 

68.  It  is  evident,  from  the  preceding  example,  that  the  frac- 
tion I  contains  unity,  or  a  whole  one,  and  ^,  and  the  reasoning, 
which  led  to  this  conclusion,  shows  also,  that  every  fractional 
expression,  of  which  the  numerator  exceeds  the  denominator, 
contains  one  or  more  units,  or  whole  ones,  and  that  these  whole 
ones  may  be  extracted  by  dividing  the  numerator  by  the  denomina- 
ior  ;  the  quotient  is  the  number  of  units  contained  in  the  fractionf 
and  the  remainder,  written  as  a  fraction,  is  that,  which  must  ac- 
company the  whole  ones. 

The  expression  y/,  for  instance,  denoting  307  parts,  of 
which  53  make  unity,  there  are,  in  the  quantity  represented  by 
this  expression,  as  many  whole  ones,  as  the  number  of  times 
53  is  contained  in  307 ;  if  the  division  be  performed,  we  shall 
obtain  5  for  the  quotient,  and  42  for  the  remainder,  these  42  are 
fifty  third  parts  of  unity  ,•  thus,  instead  of  y/,  may  be  written 

Examples  for  practice. 

Reduce  the  fraction  f  to  its  equivalent  whole  number. 

^ns.  2. 
Reduce  |  to  its  equivalent  whole  or  mixed  number,  dns.  3|. 
Reduce  *^  to  its  equivalent  whole  or  mixed  number. 

Ans.  3|. 
Reduce  \^/  to  its  equivalent  whole  or  mixed  number. 

Ans.  24/^. 
Reduce  y  to  its  equivalent  whole  or  mixed  number. 


44  Arithmetic. 

Reduce  y/  to  its  equivalent  whole  or  mixed  number. 

Ans.  Id-ij. 

69.  The  expression  S^f,  in  which  the  whole  number  is  given, 
being  composed  of  two  different  parts,  we  have  often  occasion 
to  convert  it  into  the  original  expression  ^^V,  which  is  called, 
reducing  a  whole  number  to  a  fraction. 

To  do  this,  the  whole  number  is  to  be  multiplied  by  the  denomi- 
nator of  the  accompanijing  fraction^  the  numerator  to  be  added  to 
the  product,  and  the  denominator  of  the  same  fraction  to  be  given  to 
the  sum. 

In  this  case,  the  5  whole  ones  must  be  converted  into  fifty- 
thirds,  which  is  done  by  multiplying  53  by  5,  because  each  unit 
m'Jst  contain  53  parts  ;  the  result  will  be  W*  ;  joining  this  part 
with  the  second,  ||,  the  answer  will  be  yy . 

Examples  for  practice. 

Reduce  19^  to  a  fraction.  Ms.  \*. 

Reduce  ef  to  a  fraction.  Ms.  */. 

Reduce  31-/^  to  a  fraction.  Ms.  Y/ . 

Reduce  4c^y^  to  a  fraction.  .ins.  YsV* 

70.  We  now  proceed  to  the  multiplication  of  one  fraction  by 
another. 

If,  for  instance,  |  were  to  be  multiplied  by  ^ ;  according  to  arti- 
cle 66,  the  operation  would  consist  in  dividing  |  by  5,  and  multi- 
plyingthe  result  by  4 ;  according  to  the  table  in  article  65,  the  first 
operation  is  performed  by  multiplying  3,  the  denominator  of  the 
multiplicand,  by  5  ;  and  the  second,  by  multiplying  2,  the  nume- 
ratoi'of  the  multiplicand,  by  4  ;  and  the  required  product  is  thus 
found  to  be  ■^\.       '' 

It  will  be  the  same  with  every  other  example,  and  it  must  con- 
sequently be  concluded  from  what  precedes,  that  to  obtain  the 
product  of  two  fractions^  the  two  numerators  must  be  multipliedf 
one  by  the  other t  and  under  the  product  must  be  placed  the  product  of 
the  denominators. 

Examples. 
Multiply  I  by  |.    Ms.  ^\.      Multiply  |  by  f .    Ms.  ■^. 


I  Fractions.  45 


'  Multiply  f  by  |.    Ms.  t-\.      Multiply  |»  by  ^».    Ms.  |*. 
Multiply  /t  by  1.  Ms.  ^        Multiply  1|  by  |i .     ^rts.  |-t|. 

71.  It  may  sometimes  happen  that  two  mixed  numbers,  or 
whole  numbers  joined  with  fractions,  are  to^be  multiplied,  one 
by  the  other,  as  for  instance,  S^.  by  4|.  The  most  simple  mode 
of  obtaining  the  product  is,  to  reduce  the  whole  numbers  to  frac- 
tions by  the  process  ili  article  69  ;  the  two  factors  will  then  be 
expressed  by  V  and  */'  aiid  their  product,  by  ^|f  *  or  18^f ,  by 
extracting  the  whole  ones  (68). 

72.  The  name  fractions  of  fractions  is  sometimes  given  to  the 
product  of  several  fractions  ;  in  this  sense  we  say,  |  of  -J  This 
expression  denotes  |  of  the  quantity  represented  by  |  of  the 
01  iglnal  unit,  and  taken  in  its  stead  for  unity.  These  two  frac- 
tions are  reduced  to  one  by  multiplication  (70),  and  the 
result,  j^j,  expresses  the  value  of  the  quantity  required,  with 
relation  to  the  original  unit ;  that  is,  I  of  the  quantity  rep- 
resened  by  4  of  unity  is  equivalent  to  ^^  of  unity.  If  it  were 
required  to  take  ^  of  this  result,  it  would  amount  to  taking  ^  of 
I  of  ^.  and  these  fractions,  reduced  to  one,  would  give  ^'/t  ^*^^ 
the  value  of  the  quantity  sought,  with  relation  to  the  original 
unit. 

73.  The  word  contain,  in  its  strict  sense,  is  not  more  proper  in 
the  different  cases  presented  by  division,  than  the  word  repeat  in 
those  presented  by  multiplication  ;  for  it  cannot  be  said  that  the 
dividend  contains  the  divisor,  when  it  is  less  than  the  latter ;  the 
expression  is  generally  used,  but  only  by  analogy  and  extension. 

To  generalize  division,  the  dividend  must  be  considered  as  hav- 
ing the  same  relation  to  the  quotientf  that  the  divisor  has  to  unity, 
because  the  divisor  and  quotient  are  the  two  factors  of  the 
dividend  (36) .  This  consideration  is  conformable  to  every 
case  that  division  can  present.  When,  for  instance,  the 
divisor  is  5,  the  dividend  is  equal  to  5  times  the  quotient,  and, 
consequently,  this  last  is  the  fifth  part  of  the  dividend.  If  the 
divisor  be  a  fraction,  ^  for  instance,  the  dividend  cannot  be  but 
half  of  the  quotient,  or  the  latter  must  be  double  the  former. 

The  definition,  just  given,  easily  suggests  the  mode  of 
proceeding,  when  the  divisor  is  a  fraction.    Let  us    take,  for 


46  Arithmetic. 

example,  |.  In  this  case  the  dividend  ought  to  he  only  4  of  the 
quotient ;  but  |  being  1  of  4,  we  shall  hare  -J^  of  the  quotient,  by 
taicing  ^  of  the  dividend,  or  dividing  it  by  4.  Thus  knowing  | 
of  the  quotient,  we  have  only  to  take  it  5  times,  or  multiply  it 
by  5,  tf)  obtain  the  quotient.  In  this  operation  the  dividend  is 
divided  by  4  and  multiplied  by  5,  which  is  the  same  as  taking 
f  of  the  dividend,  or  multiplying  it  by  |,  which  fraction  is  no 
other  than  the  divisor  inverted. 

This  example  shows,  that,  in  general,  to  divide  any  nuir^er  by 
afradiorif  it  must  be  viultiplied  by  the  fraction  inverted. 

For  instance,  let  it  be  required  to  divide  9  by  | ;  this  will  be 
done  by  multiplying  it  by  4*  and  the  quotient  will  be  found  to  be 
^-^  or  12.  Also  13  divided  by  4  will  be  the  same  as  13  multi- 
plied by  |,  or  y.  The  required  quotient  will  be  18|,  by  ex- 
tracting the  whole  ones  (68). 

It  is  evident  that,  whenever  the  numerator  of  the  divisor  is 
less  than  the  denominator,  the  quotient  will  exceed  the  dividend, 
because  the  divisor  in  that  case,  being  less  than  unity,  must  be 
contained  in  the  dividend  a  greater  number  of  times,  than  unity 
is,  which,  taken  for  a  divisor,  always  gives  a  quotient  exactly 
the  same  as  the  dividend. 

74.  When  the  dividend  is  afraction^  the  operation  must  be  per- 
formed by  midtiplying  the  dividend  by  the  divisor  inverted  (70). 

Let  it  be  required  to  divide  J  by  | ;  according  to  the  preced- 
ing article,  ^  must  be  multiplied  by  |.  which  gives  ^^. 

It  is  evident,  that  the  above  operation  may  be  enunciated  thus ; 
To  divide  one  fraction  by  another  ^  the  numerator  of  the  first  must 
be  multiplied  by  the  denominator  of  the  second,  and  the  denominator 
of  the  first,  by  the  numerator  of  the  second. 

If  there  be  whole  numbers  joined  to  the  given  fractions,  they 
must  be  reduced  to  fractions,  and  the  above  rule  applied  to  the 
results. 

Examples. 

Divide  7i  by -I-.  Ms.  \'. 
Divide  2|  by  S^.  Ms.  ||. 
Divide  %?  by  ^%.  Ms.  42. 
Divide  ±^  by  44.     Ms.  1. 


Divide  9  by  |. 

Ms. 

V. 

Divide  I8by  f. 

Ms. 

15. 

Divide  |  by  ^. 

Ms. 

A- 

Divide  -1 «  by  ^%. 

Ms. 

ih 

Fractions.  At 

'  75.  It  is  important  to  observe,  that  any  division,  whether  it 
can  be  performed  in  whole  numbers  or  not,  may  be  indicated  by 
a  fractional  expression ;  \^ ,  for  instance,  expresses  evidently 
the  quotient  of  56  by  3,  as  well  as  12,  for|  being  contained  three 
times  in  unity,  y  will  be  contained  3  times  in  36  units,  as  the 
quotient  of  36  by  3  must  be. 

76.  It  may  seem  preposterous  to  treat  of  the  multiplication  and 
division  of  fractions  before  having  said  any  thing  of  the  manner 
of  adding  and  subtracting  them  ;  but  tliis  order  has  been  follow- 
ed, because  multiplication  and  division  follow  as  the  imme- 
diate consequences  of  the  remark  given  in  the  table  of  arti- 
cle 55 f  but  addition  and  subtraction  require  some  previous 
preparation.  It  is,  besides,  by  no  means  surprising,  that  it 
should  be  more  easy  to  multiply  and  divide  fractions,  than  to  add 
and  subtract  them,  since  they  are  derived  from  division,  which 
is  so  nearly  related  to  multiplication.  There  will  be  many  op- 
portunities, in  what  follows,  of  becoming  convinced  of  this  truth ; 
that  operations  to  be  performed  on  quantities  are  so  much  the 
more  easy,  as  they  approach  nearer  to  the  origin  of  these  quan- 
tities. We  will  now  proceed  to  the  addition  and  subtraction  of 
fractions. 

77.  When  the  fractions  on  which  these  operations  are  to  be 
performed  have  the  same  denominator,  as  they  contain  none  but 
parts  of  the  same  denomination,  and  consequently  of  the  same 
magnitude  or  value,  they  can  be  added  or  subtracted  in  the  same 
manner  as  whole  numbers,  care  being  taken  to  mark,  in  the  re- 
sult, the  denomination  of  the  parts,  of  which  it  is  composed. 

It  is  indeed  very  plain,  that  -/^  and  -fL  make  ^*j,  as  2  quan- 
tities and  3  quantities,  of  the  same  kind,  make  5  of  that  kind, 
whatever  it  may  be. 

Also,  the  difference  between  |  and  |  is  |,  as  the  difference  be- 
tween 3  quantities  and  8  quantities,  of  the  same  kind,  is  5  of  that 
kind,  whatever  it  may  be.  Hence  it  must  be  concluded,  that,  to 
add  or  subtract  fractions^  having  the  same  denominator^  the  sum  or 
difference  of  their  numerators  must  be  taken,  and  the  common  de- 
nominator written  under  the  result.  > 

78.  When  the  given  fractions  have  different  denominators,  it 


48  Arithmetic, 

is  impossible  to  add  together,  or  subtract,  one  from  the  other, 
the  parts  of  which  they  are  composed,  because  these  parts 
are  of  different  magnitudes ;  but  to  obviate  this  difficulty,  the 
fractions  are  made  to  undergo  a  change,  which  brings  them  to 
parts  of  the  same  magnitude,  by  giving  them  a  common  denomi- 
nator. 

For  instance,  let  the  fractions  be  |  and  *  ;  if  each  term  of  the 
first  be  multijdied  by  5,  tbe  denominator  of  the  second,  the  first 
will  be  changed  into  -Jt  ,*  and  if  each  term  of  the  second  be  mul- 
tiplied by  3,  the  denominator  of  the  first,  the  second  will  be 
changed  into  [| ;  thus  too  new  expressions  w  ill  be  formed,  hav- 
ing the  same  value  as  the  given  fractions  (56). 

This  operation,  necessary  for  comparing  the  respective  mag- 
nitudes of  two  fractions,  consists  simply  in  finding,  to  express 
them,  parts  of  an  unit  suffii  iently  small  to  be  contained  exactly 
in  each  of  those  which  form  the  given  fractions.  It  is  plain,  in 
the  above  example,  that  the  fifteenth  part  of  an  unit  will  exactly 
measure  |,  and  ^  of  this  unit,  because  ^  contains  five  15^,  and 
^  contains  three  1 5^.  The  process,  applied  to  the  fi-actions  f 
and  |,  will  admit  of  being  applied  to  any  others. 

In  general,  to  reduce  any  two  fractions  to  the  same  denominator, 
the  two  terms  of  each  of  them  must  be  multiplied  by  the  denominator 
of  the  other. 

79.  Any  number  of  fractions  are  reduced  to  a  common  denomina- 
tor^  by  mnltphjing  the  two  terms  of  each  by  the  product  of  the  denom- 
inators of  all  the  others;  for  it  is  plain  that  the  new  denominators 
are  all  the  same,  since  each  one  is  the  product  of  all  the  original 
denominators,  and  that  the  new  fractions  have  the  same  value  as 
the  former  ones,  since  nothing  has  been  done  except  multiplying 
each  term  of  these  by  the  same  number  {56). 

Examples. 

Reduce  |  and  |  to  a  common  denominator.        Ms.  ||^,  |^. 
Reduce  j\  and  |^  to  a  common  denominator.      Ans.  fi.  |^. 


Reduce  -j^^,  |,  4  and  |  to  a  common  denominator. 


Fractions.  4.9 

The  preceding  rule  conducts  us,  in  all  cases,  to  the  proposed 
end  J  but  when  the  denominators  of  the  fractions  in  question  are 
not  prime  to  each  other,  there  is  a  common  denominator  more 
simple  than  that  which  is  thus  obtained,  and  which  may  be 
shown  to  result  from  considerations  analogous  to  those  given  in 
the  preceding  articles.  If,  for  instance,  the  fractions  were  |,  |, 
f ,  Y»  as  nothing  more  is  required,  for  reducing  them  to  a  com- 
mon denominator,  than  to  divide  unity  into  parts,  which  njiall  be 
exactly  contained  in  those  of  which  these  fractions  consist,  it  will 
be  sufficient  to  find  the  smallest  number,  which  can  be  exactly 
divided  by  each  of  their  denominators,  3,  4,  6,  8  ;  and  this  will 
be  discovered  by  trying  to  divide  the  multiples  of  3  by  4,  6,  8  j 
which  does  not  succeed  until  we  come  to  24,  when  we  have  only 
to  change  the  given  fractions  into  24'^"  of  an  unit. 

To  perform  this  operation  we  must  ascertain  successively  hovr 
many  times  the  denominators,  3,  4,  6  and  8,  are  contained  in 
24,  and  the  quotients  will  be  the  numbers,  by  which  each  term 
of  the  respective  fractions  must  be  multiplied,  to  be  reduced  to 
the  common  denominator,  24.  It  will  thus  be  found,  that  each 
term  of  f  must  be  multiplied  by  8,  each  term  of  |  by  6,  each 
term  of  f  by  4,  and  each  term  of  |  by  3  j  the  fractions  will  then 
become  i|,  i|,  |^,  |l. 

Algebra  will  furnish  the  means  of  facilitating  the  application 
of  this  process. 

80.  By  reducing  fractions  to  the  same  denominator,  they  may 
be  added  and  subtr-acted  as  in  article  77. 

81.  When  there  are  at  the  same  time  both  whole  numbers  and 
fractions,  the  whole  numbers,  if  they  stand  alone,  must  be  con- 
verted into  fractions  of  the  same  denomination  as  those,  which 
are  to  be  added  to  them,  or  subtracted  from  them  ;  and  if  the 
whole  numbers  are  accompanied  with  fractions,  they  must  be 
reduced  to  the  same  denominator  with  these  fractions. 

It  is  thus,  that  the  addition  of  4  units  and  |  changes  itself  into 
the  addition  of  y  and  |,  and  gives  for  the  result  Y- 

To  add  3^  to  5|,  the  whole  numbci-s  must  be  reduced  to  frac- 
tions, of  the  same  denomination  as  those  which  accompany  them, 
which  reduction  gives  ^^  and  V  ?  with  these  results  the  sum  is 


50  Jirithmetic, 

found  to  be  y/,  or  8f|.  If,  lastly,  |  were  to  be  subtracted  from 
3^,  the  operation  would  be  reduced  to  taking  |  from  y ,  and  the 
remainder  would  be  ||. 


additio7t  of  fractions. 

Ms.  II, 


Ms.  1^,  or  1. 
Ms.  II. 
Ms.  ±1. 

Ms.    S-g-^-y. 

Ms.  12^\. 
^ns.  8|. 


Examples  in  subtraction  of  fractions. 

From  I    take  ^.    ^ns.  ^.      From  5|  take  2|.      Ans.  2|. 

From  I    take  |.    w3/is.  -jV*     From  8|  take  4|.      ,3ns.  4^'^. 

From  II  take  /^.  wJtis.  |.       From  31  take  2A°.     ,/2/is.  If. 

82.  The  rule  given,  for  the  reduction  of  fractions  to  a  com- 
mon denominator  supposes,  that  a  product  resulting  from  the 
successive  multiplication  of  several  numbers  into  each  other, 
does  not  vary,  in  whatever  order  these  multiplications  may  b« 
performed ;  this  truth,  though  almost  always  considered  as  self- 
evident,  needs  to  be  proved. 

We  shall  begin  with  showing,  that  to  multiply  one  number  by 
the  product  of  two  others  is  the  same  thing  as  to  multiply  it  at 
first  by  one  of  them,  and  then  to  multiply  that  product  by  the 
other.  For  instance,  instead  of  multiplying  3  by  S5,  the  pro- 
duct of  7  and  5,  it  will  be  the  same  thing,  if  we  multiply  3  by  5, 
and  then  that  product  by  7.  The  proposition  will  be  evident,  if, 
instead  of  3,  we  take  an  unit;  for  1,  multiplied  by  5,  gives  5, 
and  the  product  of  5  by  7  is  35,  as  well  as  the  product  of  1  by 
35  ;  but  3,  or  any  other  number,  being  only  an  assemblage  of 
several  units,  the  same  property  will  belong  to  it,  as  to  each  of 
the  units  of  which  it  consists  ;  that  is,  the  product  of  3  by  5  and 
by  7,  obtained  in  either  way,  being  the  triple  of  the  results 
given  by  unity,  when  multiplied  by  5  and  7,  must  necessarily  be 
the  same.     It  may  be  proved  in  the  same  manner,  that  were  it 


Decimal  Fradions.  51 

rcquired  to  multiply  3  by  the  product  of  5,  7  and  9,  it  would 
consist  in  multiplying  3  by  5,  then  this  product  by  7,  and  the 
result  by  9,  and  so  on,  whatever  might  be  the  number  of  factors. 

To  represent  in  a  shorter  manner  several  successive  multipli- 
cations, as  of  the  numbers  3,  5,  and  7  into  each  other,  we  shall 
write  3  by  5  by  7. 

This  being  laid  down,  in  the  product  3  by  6,  the  order  of  the 
factors,  3  and  5  (27),  may  be  changed,  and  the  same  product 
obtained.  Hence  it  directly  follows,  that  5  by  3  by  f  is  the  same 
as  3  by  5  by  7. 

The  order  of  the  factors  3  and  7,  in  the  product  5  by  3  by  7, 
may  also  be  changed,  because  this  product  is  equivalent  to  5 
multiplied  by  the  product  of  the  numbers  3  and  7 ;  thus  we  have 
in  the  expression  5  by  7  by  3,  the  same  product  as  the  preceding. 

By  bringing  together  the  three  arrangements, 
3  by  5  by  7 
5  by  3  by  7 
5  by  7  by  3, 
we  see  that  the  factor  3  is  found  successively,  the  first,  the  second, 
and  the  third,  and  that  the  same  may  take  place  vvitli  res])ect 
to  either  of  the  others.     Fn>m  this  example,  in  which  the  par- 
ticular value  of  each  number  has  not  been  considered,  it  must 
be  evident,  that  a  product  of  three  factors  does  not  vary,  what- 
ever may  be  the  order  in  which  they  ai-e  multiplied. 

If  the  question  were  concerning  the  product  of  four  factors, 
such  as  3  by  5  by  7  by  9,  we  might,  according  to  what  has  been 
said,  arrange,  as  we  pleased,  the  three  first  or  the  three  last,  and 
thus  make  any  one  of  the  factors  pass  through  all  the  places. 
Considering  then  one  of  the  new  arrangements,  for  instance,  this 
5  by  7  by  3  by  9,  we  might  invert  the  order  of  the  two  last  fac- 
tors, which  would  give  5  by  7  by  9  by  3,  and  would  put  3  in  the 
last  place.  This  reasoning  may  be  extended  without  difficulty 
to  any  number  of  factors  whatever. 

Decimal  Fractions. 

83.  Although  we  can,  by  the  preceding  rules,  apply  to  frac- 
tions, in  all  cases;  the  four  fundamental  operations  of  arithmetic^ 


52  Arithmetic. 

yet  it  must  have  been  long  since  perceived,  that,  if  the  different 
subdivisions  of  a  unit,  employed  for  measurin.a;  quantities  smaller 
than  this  unit,  had  been  subjected  to  a  common  law  of  decrease,  the 
calculus  of  fractions  would  have  been  much  more  convenient,  on 
account  of  the  facility  with  which  we  might  convert  one  into 
another.  By  making  this  law  of  decrease  conform  to  the  basis 
of  our  system  of  numeration,  we  have  given  to  the  calculus  the 
greatest  degree  of  simplicity,  of  which  it  is  capable. 

We  have  seen  in  article  5,  that  each  of  the  collections  of  units, 
contained  in  a  number,  is  composed  often  units  of  the  preceding 
order,  as  the  ten  consists  of  simple  units  ;  but  there  is  nothing 
to  prevent  our  regarding  this  simple  unit,  as  containing  ten 
parts,  of  which  each  one  shall  be  a  tenth  ;  the  tenth  as  containing 
ten  parts,  of  which  each  one  shall  be  a  hundredth  of  unity,  the 
hundredth  as  containing  ten  parts,  of  which  each  one  shall  be  a 
thousandth  of  unity,  and  so  on. 

Proceeding  thus,  we  may  form  quantities  as  small  as  we 
please,  by  means  of  which  it  will  be  possible  to  measure  any 
quantities,  however  minute.  These  fractions,  which  are  called 
decimals^  because  they  are  con  posed  of  parts  of  unity,  that  be- 
come continually  ten  times  smaller,  as  they  depart  further  from 
unity,  may  be  converted,  one  into  the  other,  in  the  same  manner 
as  tetis,  hundreds,  thousands,  &c.  are  converted  into  units ;  thus, 
the  unit  being  equivalent  to  10  tenths, 
the  tenth  10  hundredths, 

the  hundredth  10  thousandths, 

it  follows,  that  the  tenth  is  equivalent  to  10  times  10  thousandths, 
or  iOO  thousandths. 

For  instance,  2  tenths,  3  hundredths  and  4  thousandths  will 
be  equivalent  to  234  thousandths,  as  2  hundreds,  3  tens  and  4 
units  malce  234  units ;  and  what  is  here  said  may  be  applied 
universally,  since  the  subordination  of  the  parts  of  unity  is  like 
that  of  the  different  orders  of  units. 

84.  According  to  this  remark,  we  can,  by  means  of  figures, 
write  decimal  fractions  in  the  same  manner  as  whole  numbers, 
since  by  the  nature  of  our  numeration,  which  makes  the  value  of 
a  figure,  placed  on  the  right  of  another,  ten  times  smaller,  tenth's 


Decimal  Fractions.  5S 

naturally  take  their  place  on  the  right  of  units,  then  hundredths 
on  the  right  of  tenths,  and  so  on  ;  but,  that  the  figures  express- 
ing decimal  parts  may  not  be  confounded  with  those  expressing 
whole  units,  a  commaf  is  placed  on  the  right  of  units.  To  ex- 
press, for  instance,  34  units  and  27  hundredths,  we  write  34,27. 
If  there  he  no  units,  their  place  is  suj)plied  by  a  cipher,  and  the 
same  is  done  for  all  tlie  decimal  parts,  which  may  be  wanting 
between  those  enunciated  in  the  given  number. 
Thus  19  hundredths  are  written  0,19, 
304  thousandths  0,304  , 

3  thousandths  0,003 . 

85.  If  the  expressions  for  the  above  decimal  fractions  be  com- 
pared with  the  following,  -jW ,  -^y>^^ ,  -j-^^^-^  ,  drawn  from  the 
general  manner  of  representing  a,  fraction,  it  will  be  seen,  that 
to  represent  in  an  entire  form  a  decimal  fraction^  written  as  a  vul- 
gar fraction,  the  numerator  of  the  fraction  must  be  taken  as  it  is, 
and  placed  after  the  comma  in  such  a  manner,  tliat  it  may  have  as 
manyfgures  as  there  are  ciphers  after  the  unit  in  the  denominator. 

Reciprocally,  to  reduce  a  decimal  fraction,  given  in  the  form  of 
a  whole  number,  to  that  of  a  vulgar  fraction,  the  figures,  that  it 
contains,  must  receive,  for  a  denominator,  an  unit  followed  by  as 
many  ciphers,  as  there  are  figures  after  the  comma. 

Thus  the  fractions,  0,56  ,  0,036  ,  are  changed  into  //^  and 

86 

86.  Jin  expression,  in  figures,  of  numbers  containing  decimal 
parts,  is  read  by  enunciating,  first,  the  figures  placed  on  the  left  of 
the  point,  then  those  on  the  right,  adding  to  the  last  figure  of  the 
latter  the  denomination  of  the  parts,  which  it  represents. 

The  number  26,736  is  read  26  and  736  thousandths ; 
the  number  0,0675  is  read  673  ten  thousandths, 
and  0,0000673  is  read  673  ten  millionths. 


t  In  English  books  on  mathematics,  and  in  those  that  have  been 
written  in  the  United  States,  decimals  are  usually  denoted  by  a 
point,  thus  0-19  J  but  the  comma  is  on  the  whole  in  the  most  general 
use ;  it  is  accordingly  adopted  in  this  and  the  subsequent  treatises 
to  be  published  at  Cambridge. 


54  Mthmetic. 

87.  As  decimal  figures  take  their  value  entirely  from  their 
position  relative  to  the  comma,  it  is  of  no  consequence  whether 
we  write  or  omit  any  number  of  ciphers  on  their  ri.i;ht.  For 
instance,  0,5  is  the  same  as  0,50  j  and  0,784  is  the  sa.ne  as 
0,78400  ;  for,  in  the  first  instance,  the  number,  which  exjjresscs 
the  decimal  fraction,  becomes  by  the  addition  of  a  0  ten  times 
greater,  but  the  parts  become  hundredths,  and  consequently 
on  this  account  are  ten  times  less  than  before ;  in  the  second 
instance,  the  number,  which  expresses  the  fraction,  becomes  a 
hundred  times  greater  than  before,  but  the  parts  become  hun- 
dred thousandths,  and,  consequently  are  a  hundred  times  smaller 
than  before.  Tliis  transformation,  then,  becomes  the  same  as 
that  which  takes  place  with  respect  to  a  vulgar  fraction,  when 
each  of  its  terms  is  multiplied  by  the  same  number ;  and  if  the 
ciphers  be  suppressed,  it  is  the  same  as  dividing  them  by  the 
same  number. 

88.  The  addition  of  decimal  fractions  and  numbers  accompa- 
nyi!>g  them,  needs  no  other  rule  than  that  given  for  whole  num- 
bers, since  the  decimal  parts  are  made  up  one  from  the  other, 
ascetidingfrom  right  to  left,  in  the  same  manner  as  whole  units. 

For  instance,  let  there  be  the  numbers  0,56,  0,003,  0,958; 
disposing  them  as  follows, 

0,56 

0,003 

0,958 


Sum  1,521 

we  find,  by  the  rule  of  article  12,  that  their  sum  is  1,521. 

Again,  let  there  be  the  numbers  19,35  ,  0,3  ,  48,5  ,  and  110,02, 
which  contain  also  whole  units,  they  will  be  disposed  thus  ; 
19,35 

0,3 

48,5 

110,02 


Sum  178,17 

and  their  sum  will  be  178,17. 
In  general,  the  addition  of  decimal  numbers  is  performed  like 


Decimal  Fractions.  tlfi 

that  of  whole  numherst  care  being  taken  to  place  the  comma  in  the 
sum,  directly  under  the  commas  in  the  numbers  to  be  added. 

Examples  for  practice. 

Add  4,003,  54,9,  3,21,  6,7203.  ^ns.  68,8333. 

Add  409,903,  107,7842,  6,1043,  10,2974.     dns.  534,0889. 

Add  427,  603,04,  210,l5,  3,364,  ,021.  ,iiis.  1243,575. 

89.  The  rules  prescribed  for  the  suhtraction  of  whole  num- 
bers, apply  also,  as  will  be  seen,  to  decimals.  For  instance,  let 
0,3697  be  taken  from  0,62  ,•  it  must  first  be  observed,  that  the 
second  number,  which  contains  only  hundredths,  while  the 
other  contains  ten  thousandths,  can  be  converted  into  ten  thou- 
sandths by  placing  two  ciphers  on  its  right  (87),  which  changes 
it  into  0,6200. 

The  operation  will  then  be  arranged  thus ; 
0,6200 
0,3697 


Difference       0,2503 
and,  according  to  the  rule  of  article  17,  the  difference  will  be 
0,2503. 

Again,  let  7,364  be  taken  from  9,1457  ;  the  operation  being 
disposed  thus ; 

9,1457 
7,3640 


Difference      1,7817 
the  above  difference  is  found.  It  would  have  been  just  as  well  if  no 
cipher  had  been  placed  at  the  end  of  the  number  to  be  subtracted, 
provided  its  different  figures  had  been  placed  under  the  corres- 
ponding orders  of  units  or  parts,  in  the  upper. 

In  general,  the  subtraction  of  decimal  numbers  is  performed  Hke 
that  of  whole  numbers^  provided  that  the  number  of  decimal  figures, 
in  the  two  given  numbers^  be  made  alike,  by  writing  on  the  right 
of  that,  which  has  the  least,  as  many  ciphers  as  are  7iecessary  ;  and 
that  the  comma  in  the  difference  is  put  directly  under  those  of  the 
given  numbers. 


56  Arithmetic. 

Examples  for  practice. 

From  304,567  take  158,632.  Ms.  145,935. 

From  215,003  take  1,1034.  .5ms.  213,8996. 

From  1  take  ,9993.  Ans.  0,0007. 

From  68,8333  take  ,00042.  Ans.  68,83288. 

The  methods  of  proving  addition  and  subtraction  of  decimals 
are  the  same  as  those  for  the  addition  and  subtraction  of  whole 
numbers. 

90.  As  the  comma  separates  the  collections  of  entire  units 
from  the  decimal  parts,  by  altering  its  place  we  necessarily 
change  the  value  of  the  whole.  By  moving  it  towards  the  right, 
figures,  which  were  contained  in  the  fractional  part,  are  made  to 
pass  into  that  of  whole  numbers,  and  consequently  the  value  of 
the  given  number  is  increased.  On  the  contrary,  by  moving  the 
comma  towards  the  left,  figures,  which  were  contained  in  the  part 
of  whole  numbers,  are  made  to  pass  into  that  of  fractions,  and 
consequently  the  value  of  the  given  number  is  diminished. 

The  first  change  makes  the  given  number,  ten,  a  hundred,  a 
thousand,  &c.  times  greater  than  before,  according  as  the  comma 
is  removed  one,  two,  three,  &c.  places  towards  the  right,  because 
for  each  place  that  the  comma  is  thus  removed,  all  the  figures 
advance  with  respect  to  this  comma  one  place  towards  the  left, 
and  consequently  assume  a  value  ten  times  greater  than  they  had 
before. 

If,  for  example,  in  the  number  134,28 ,  the  point  be  placed 
between  the  2  and  the  8,  we  shall  have  1342,8,  the  hundreds 
will  have  become  thousands,  the  tens  hundreds,  the  units  tens, 
the  tenths  units,  and  the  hundredths  tenths.  Every  pai't  of  the 
number  having  thus  become  ten  times  greater,  the  result  is  the 
same  as  if  it  had  been  multijdied  by  ten. 

The  second  change  makes  the  giv  ( n  number  ten,  a  hundred,  a 
thousand,  &c.  times  smaller  than  it  was  before,  according  as  the 
comma  is  removed  one,  two,  three,  &c.  places  towards  the  left, 
because  for  each  place  that  the  comma  is  thus  removed,  all  the 
figures  recede,  with  respect  to  this  comma,  one  place  further  to 
the  right,  and  consequently  have  a  value  ten  times  less  than 
they  had  before. 


Beeimal  Fractions,  S7 

If,  in  the  number  134,26,  the  point  be  placed  between  the  3 
and  4,  we  shall  have  13,428 ;  the  hundreds  will  become  tens, 
the  tens  units,  the  units  tenths,  the  tenths  hundredths,  and 
the  hundredths  thousandths ;  every  part  of  the  number  having 
thus  becoirie  ten  times  smaller,  the  result  is  the  same  as  if  a 
tenth  part  of  it  had  been  taken,  or  as  if  it  had  been  divided  by  ten. 

91.  From  what  has  been  said,  it  will  be  easy  to  perceive  the 
advantage,  which  decimal  fractions  have  over  vulgar  fractions  j 
all  the  multiplications  and  divisions,  which  are  performed  by 
the  denominator  of  the  latter,  are  performed  with  respect  to  the 
former,  by  the  addition  or  suppression  of  a  number  of  ciphers,  or 
by  simply  changing  the  place  of  the  comma.  By  adapting  these 
modifications  to  the  theory  of  vulgar  fractions,  we  thence  imme- 
diately deduce  that  of  decimals,  and  tlie  manner  of  performing 
the  multiplication  and  division  of  them  j  but  we  can  also  arrive 
at  this  theory  directly  by  the  following  considerations. 

Let  us  first  suppose  only  the  multiplicand  to  have  decimal 
figures.  If  the  comma  be  taken  away,  it  will  become  ten,  a 
hundred,  a  thousand,  &c.  times  greater,  according  to  the  num- 
ber of  decimal  figures ;  and  in  this  case  the  product  given  by 
multiplication  will  be  a  like  number  of  times  greater  than  the 
one  required ;  the  latter  will  then  be  obtained  by  dividing  the 
former  by  ten,  a  hundred,  a  thousand,  &c.  wliich  may  be  done  by 
separating  on  the  right  (90)  as  many  decimal  figures,  as  there 
are  in  the  multiplicand. 

If,  for  instance,  34,137  were  to  be  multiplied  by  9,  we  must 
first  find  the  product  of  34137  by  9,  which  will  be  307233  ;  and, 
since  taking  away  the  comma  renders  the  multiplicand  a  thou- 
sand times  greater,  we  must  divide  this  product  by  a  thousand, 
or  separate  by  a  comma,  its  three  last  figures  on  the  right  -,  we 
shall  thus  have  307,233. 

In  general,  to  multiply^  hy  a  whole  numhery  a  number  accompa- 
nied l)y  decimals^  the  comma  must  he  taken  away  from  the  multi- 
plicand, and  as  many  figures  separated  for  decimals,  on  the  righ4 
»f  the  product,  as  are  contained  in  the  multiplicand. 


58  Arithmetic, 

Examples  for  practice, 

.    Multiply  231,415  by  8.  Am.  1851,320. 

Multiply  32,1509  by  15.  Am.  482,26j5. 

Multiply     ,840     by  840.  Am.  705,600. 

Multiply    1,236    by  13.  Am.  16,068. 

92.  When  the  multiplier  contains  decimal  figures  by  sup- 
pressing the  comma,  it  is  made  ten,  a  hundred,  a  thousand,  &c. 
times  greater  accoiding  to  the  number  of  decimal  figures.  If 
used  in  this  state,  it  will  evidently  give  a  product,  ten,  a  hun- 
dred, a  thousand,  &c.  times  greater  than  that  wliich  is  required, 
and  consequently  the  true  product  will  be  obtained  by  dividing 
by  one  of  these  numbers,  that  is,  by  separating,  on  the  right  of 
it,  as  many  decimal  figures  as  there  are  in  the  multiplier,  or  by 
removing  the  comma  a  like  number  of  places  towards  the  left  (90), 
in  case  it  previously  existed  in  the  ])roduct  on  account  of  de- 
cimals in  the  multiplicand.  For  instance,  let  172,84  be  mul- 
tiplied by  36,003  ;  taking  away  the  comma  in  the  multiplier  only, 
we  shall  have,  according  to  the  preceding  article,  the  product 
6222758,52;  but,  the  multiplier  being  rendered  a  thousand  times 
too  great,  we  must  divide  this  product  by  a  thousand,  or  remove 
the  comma  three  places  towards  the  left,  and  the  required  pro- 
duct will  then  be  6222,75852,  in  which  there  must  necessarily  be 
as  many  decimal  figures  as  there  are  in  both  multiplicand  and 
multiplier. 

In  general,  to  multiply  one  hy  the  other^  txvo  numbers  accompa- 
nied by  decimals^  the  comma  must  be  taken  away  from  both,  and  as 
many  figures  separated  for  decimals,  on  the  right  of  the  prodiict, 
as  there  are  in  both  the  factors. 

In  some  cases  it  is  necessary  to  put  one  or  more  ciphers  on  the 
left  of  the  product,  to  give  the  number  of  decimal  figures  requir- 
ed by  the  above  rule.  If,  for  example,  0,624  be  multiplied  by 
0,003  ;  in  forming  at  first  the  product  of  624  by  3,  we  shall  have 
the  number  1872,  containing  but  4  figures,  and  as  6  figures  must 
be  separated  for  decimals,  it  cannot  be  done  except  by  placing 
on  the  left  three  ciphers,  one  of  which  must  occupy  the  place  of 
units,  which  will  make  0,001872, 


Jkdmal  Fractions.  t^ 

'  Examples  for  practice. 

Multiply  223,86  by  2,500.  Ms,  559,65000. 

Multiply  35,640  by  26,18.  Arts.  933,05520. 

Multiply  8,4960  by  2,618.  Ans,  22,2425280. 

Multiply  ,5236    by  ,280^.  Ms.  0,14702688. 

Multiply  ,1 1785  by  ,27.  Ms.  0,0318195. 

93.  It  is  evident  (36),  that  the  quotient  of  two  numbers  does 
not  depend  on  the  absolute  maj^nitude  of  their  units,  provided 
that  this  be  tlie  same  in  each  ;  if  then,  it  be  required  to  divide 
451,49  by  13,  we  should  observe  that  the  former  amounts  to 
45149  hundredths,  and  the  latter  to  1300  hundredths,  and  that 
these  last  numbers  ought  to  give  the  same  quotient,  as  if  they 
expressed  units  We  shall  thus  be  led  to  suppress  the  point  in 
the  first  number,  and  to  put  two  ciphers  at  the  end  of  the  second, 
and  then  we  shall  only  have  to  divide  45149  by  1300,  the  quo- 
tient of  which  division  will  be  34  —^W. 

Hence  we  conclude,  that,  to  divtde,  by  a  whale  number,  a  num- 
ber accompanied  by  decimal  Jigures,  the  comma  in  the  dividend  must 
be  taken  away,  and  as  many  ciphers  placed  at  the  end  of  the  divisor, 
as  the  dividend  contains  decimal  figures,  and  no  alteration  in  the 
qaotient  will  be  necessary. 

94.  When  both  dividend  and  divisor  are  accompanied  by  deci- 
mal figures,  we  must,  before  taking  away  the  comma,  reduce 
them  to  decimals  of  the  same  order,  by  placing  at  the  end  of  that 
number,  which  has  the  fewest  decimal  figures,  as  many  ciphers 
as  will  make  it  terminate  at  the  same  place  of  decimals  as  the 
other,  because  then  the  suppression  of  the  comma  renders  both  the 
same  number  of  times  greater. 

For  instance,  let  315,432  be  divided  by  23,4,  this  last  must  be 
changed  into  23,400,  and  then  315432  must  be  divided  by  23400  ; 
the  quotient  will  be  I5im|.. 

Thus,  to  divide  one  by  the  other,  two  numbers  accompanied  by 
decimal  fgures,  the  number  of  decimal  figures  in  the  divisor  and 
dividend  must  be  made  equal,  by  annexing  to  the  one,  that  has  the 
least,  as  many  ciphers  as  are  necessary  ;  the  point  must  then  be  sup- 
pressed in  each,  and  the  quotient  will  require  no  alteration. 

95.  As  we  have  recoui'se  to  decimals  only  to  avoid  the  neces- 


69  Arithmetic. 

sity  of  employing  vulgar  fractions,  it  is  natural  to  make  use  of 
decimals  for  approximating  quotients  that  cannot  be  obtained 
exactly,  which  is  done  by  converting  the  remainder  into  tenths, 
hundredths,  thousandths,  &c.  so  that  it  may  contain  the  divisor ; 
as  may  be  seen  in  the  following  example ; 


45149 
3900 

1300 

34,73 

6149 

5200 

Remainder 

949 

tentiis 

9490 

9100 

hundredths 

3900 

3900 

0 

When  we  come  to  the  remainder  949,  we  annex  a  cipher  i« 
order  to  multiply  it  by  ten,  or  to  convert  it  into  tenths ;  thus 
forming  a  new  partial  dividend,  which  contains  9490  tenths  and 
gives  for  a  quotient  7  tenths,  which  we  put  on  the  right  of  the 
units,  after  a  comma.  There  still  remains  390  tenths,  which 
we  reduce  to  hundredths  by  the  addition  of  another  cipher,  and 
form  a  second  dividend,  which  contains  3900  hundredths,  and 
gives  a  (piotient,  3  hundredths,  which  we  place  after  the  tenths. 
Here  the  operation  terminates,  and  we  have  for  the  exact  result 
34,73  hundredths.  If  a  third  remainder  had  been  left,  we  might 
have  continued  the  operation,  by  converting  this  remainder  into 
thousandths,  arid  so  on,  in  the  same  manner,  until  we  came  to 
an  exact  quotient,  or  to  a  remainder  composed  of  parts  so  small, 
that  we  might  have  considered  them  of  no  importance. 

It  is  evident,  that  we  must  always  put  a  comma,  as  in  the 
above  example,  after  the  whole  units  in  the  quotient,  to  distin- 
guish them  from  the  decimal  figures,  the  number  of  which  must 
he  equal  to  that  of  the  ciphers  successively  written  after  the 
remainders*. 

*  The  problem  above  performed  with  respect  to  decimals,  is  only 


JBedmal  Fractions. 


61 


Examples  for  practice. 

Divide  6345,925 

Divide  5673,21 

Divide  84329907 

Divide  27845,96 

Divide  200,5 

Divide  10,0 

Divide  513,2 
■    Divide  7,25406 

Divide  0,0007875 

Divide  14 

96.  The  numerator  of  a  fraction,  being  converted  into  decimal 
parts,  can  be  divided  by  the  denominator  as  in  the  preceding 
examples,  and  by  this  means  the  fraction  will  be  converted  into 
decimals.  Let  the  fraction,  for  example,  be  |,  the  operation  is 
performed  thus ; 

1 


by  54.23. 

w3m5.  117,018  &c. 

by  23,0. 

Jns.  246,660  &c. 

by  627,1. 

Ms.  134476,10  &c. 

by  9,8732. 

.ans.  2820,b581  &c. 

by  231. 

Ms.  0,0867  &c. 

by  563,0. 

Ms.  0,00177  &c. 

by  0,057. 

Ms.  9003,50  &c. 

by  957. 

Arts.  0,00758  &c. 

by  0,525. 

.5ns.  0,00150  &c. 

by  365. 

Ms.  0,038356  &c. 

20 
16 


0,125 


40 
40 


Again,  let  the  fraction  be  y^y ;  the  numerator  must  be  con- 
verted into  thousandths  before  the  division  can  begin. 


a  particular  case  of  the  following  more  general  one ;  To  find  the 
value  of  the  quotient  of  a  division^  infractions  of  a  given  denomina- 
tion ;  to  do  this,  we  convert  the  dividend  into  a  fraction  of  the  same 
denomination  by  multiplying  it  by  the  given  denominator.  Thus,  in 
order  to  find  in  fiiteenths  the  value  of  the  quotient  of  7  by  3,  we 
should  multiply  7  by  15,  and  divide  the  product,  105,  by  3,  which 
gives  thirty -five  fifteenths,  or  4f  for  tf'e  quotient  required. 


62  Jrithmetic. 


797 


40„'0 
3985 


0,0Uo018  kc. 


1500 

797 


7030 
6376 


654 

Examples  for  practice. 

Reduce  |  to  a  decimal  fraction.  Jns.  0.75. 

Reduce  |  to  a  decimal  fraction.  Jns.  0,5. 

Jus.  0,0714285  &c. 
Jm.  0.05. 
Jns.  0,333  &c. 

97.  However  far  we  may  continue  the  second  division,  exhib- 
ited above,  we  shall  never  obtain  an  exact  quotient,  because  the 
fraction  yl^  cannot,  like  |,  be  exactly  expressed  by  decimals. 

The  difference  in  the  two  cases  arises  from  this,  tliat  the  de- 
nominator of  a  fraction,  which  does  not  divide  its  numerator, 
cannot  give  an  exact  quotient,  except  it  v\ill  divide  one  of  the 
numbers  10,  100,  1000,  &c.  by  which  its  numerator  is  suc- 
cessively multiplied,  because  it  is  a  principal,  which  will  be 
found  demonstrated  in  Algebra,  that  no  number  vvill  divide  a 
product,  except  its  factors  will  divide  those  of  the  product ;  now 
the  numbers  10,  100,  1000,  &c.  being  all  formed  from  10,  the 
factors  of  which  are  2  and  5,  they  cannot  be  divided  except  by 

*  It  may  also  be  proposed  to  convert  a  given  fraction  into  a  frac- 
tion of  another  denomination,  but  smaller  than  the  first,  for  instance, 
1  into  seventeenths,  which  will  be  done  by  multiplying  3  by  17  and 
dividing  the  product  by  4.  In  this  manner  we  find  *J  seventeenths, 
or  if  and  1  of  a  seventeenth  :  but  |  of -J:^  is  equivalent  to  ^,  The 
result  tuen,  if,  is  equal  to  |,  wanting  -g\. 

This  operation  and  that  of  the  preceding  note  depend  on  the  same 
principle,  as  the  corresponding  operation  for  decimal  fractions. 


Decimal  Fractions,  6S 

numbers  formed  from  these  same  factors  ,•  8  is  among  these, 
being  the  product  of  2  by  2  by  2. 

Fractions,  the  value  of  which  cannot  be  exactly  found  by  de- 
cimals, present  in  their  approxiuiate  expression,  when  it  has 
been  carried  siuffici.'ntly  far,  a  character  which  serves  to  denote 
them  ;  this  is  the  periodical  return  of  the  same  figures. 

If  we  convert  the  fraction  ^f  into  decimals,  we  shall  find  it 

0,324324 ,  and  the  figures  3,  2,  4,  will  always  return  in 

the  same  order,  without  the  operation  ever  coming  to  an  end. 

Indeed,  as  there  can  be  no  remainder  in  these  successive 
divisions  except  one  of  the  series  of  whole  numbers  1,  2,  3,  &c. 
up  to  the  divisor,  it  necessarily  happens,  that,  when  the  number 
of  divisions  exceeds  that  of  this  series,  we  must  fall  again  upon 
some  one  of  the  preceding  remainders,  and  coLsequeutly  the 
partial  dividends  will  return  in  the  same  order.  In  the  above 
example  three  divisions  are  sufficient  to  cause  the  return  of  the 
same  figures  ;  but  six  are  necessary  for  the  fraction  I,  because 
in  this  case  we  find,  for  remainders,  the  six  numbers  which  are 
below  7,  and  the  result  is  0,1428571  ....  The  fraction  -|  leads 
only  to  0,3333 

98.  The  fractions,  which  have  for  a  denominator  any  num- 
bers of  9s,  have  no  significant  figure  in  their  periods  except  1 ; 
^   gives     0,11111  ..  . 


?-  0,010101 


T? 


1-  0,001001001 


and  so  with  the  others,  because  each  partial  division  of  the  num- 
bers 10,  100,  1000,  &c.  always  leaves  unity  for  the  remainder. 

Availing  ourselves  of  this  remark,  we  pass  easily  from  a 
periodical  decimal,  to  the  vulgar  fraction  from  which  it  is  deriv- 
ed.    We  see,  for  example,  that  0,33333 amounts  to  the 

sa,me  as  0,1111 1 multiplied  by  3,  and  as  this  last  decimal 

is  the  development  of  ^,  or  ^  reduced  to  a  decimal,  we  conclude, 
that  the  former  is  the  development  of  1  multiplied  by  3,  or  |,  or 
lastly,  1. 

When  the  period  of  the  fraction  under  consideration  consists 
of  two  figures,  we  compare  it  with  the  development  of  ^V,  and  with 
that  of -y-l-/,  when  the  period  contains  three  figures,  and  so  on. 


64  JlrithmeUc, 

If  we  had,  for  example,  0,324324,  it  is  plain  that  this  fraction 

may  be  formed  by  multiplying  0,001001 by  3^i4  j   if  we 

multiply  then  ^|.^,  of  wliich  0,001001  is  the  development,  by  324, 
we  obtain  ||^,  and  dividing  each  term  of  this  result  by  27,  we 
come  back  again  to  the  fraction  ^^. 

In  general,  the  vulgar  fraction,  from  which  a  decimal  fraction 
ariseSf  is  formed  by  writings  as  a  denominator^  under  the  number, 
which  expresses  one  period,  as  many  9s,  as  there  are  figures  in  the 
period. 

If  the  period  of  the  fraction  does  not  commence  with  the 
first  decimal  figure,  we  can  for  a  moment  change  the  place  of 
the  point,  and  put  it  immediately  before  the  first  figure  of  the 
period,  and,  beginning  with  this  figure,  find  the  value  of  the 
fraction,  as  if  those  figures  on  the  left  were  units;  nothing  then 
will  be  necessary  except  to  divide  the  result  by  10,  100,  1000, 
&c.  according  to  the  number  of  places  the  point  was  moved  to- 
wards the  right. 

For  instance,  the  fraction  0,324141,  is  first  to  be  written 
32,4141 ;  the  part  0,4141  being  equivalent  to  li,  we  shall  have 
32^.  which  is  to  be  divided  by  100,  because  the  point  was 
moved  two  places  towards  the  left ;  it  will  consequently  become 
^^-^  and  ^l^y,  or  by  reducing  the  two  parts  to  the  same  denom- 
inator, and  adding  them,  ||^|^>  a  fraction,  which  will  reproduce 
the  given  expression. 

Examples  for  practice*. 
Reduce  0,18  to  the  form  of  a  vulgar  fraction.  Jlns.  y\ 

Reduce  0,72  to  the  form  of  a  vulgar  fraction.  Ans.  ■^^. 

Reduce  0,83  to  the  form  of  a  vulgar  fraction.  Jlns.  f . 

Reduce  0,241 8  to  the  form  of  a  vulgar  fraction.        Ms.  \^%j» 
Reduce  0,275463  to  the  form  of  a  vulgar  fraction. 

qnq     3  3  9  5  3 

Reduce  0,916  to  the  form  of  a  vulgar  fraction.         Ans.  ii. 

*  In  these  examples,  the  better  to  distinguish  the  period,  a  point  is 
placed  over  it,  if  it  be  a  single  figure,  and  over  the  first  and  last 
figure,  if  it  consist  of  more  than  one. 


Tables  of  Cohif  Weight,  and  Measure.  65 

To  form  a  correct  idea  of  the  nature  of  these  fractions  it  is 
sufficient  to  consider  the  fraction  0,999.  In  trying  to  discover  its 
original  value  we  find  that  it  answei-s  to  9  divided  by  9,  that  is 
to  unity  ;  nevertheless,  at  whatever  number  of  figures  we  stop  in 
its  expression,  it  will  never  make  an  unit.  If  we  stop  at  the 
first  figure,  it  wants  -^^  of  an  unit ;  if  at  the  second,  it  wants 
j^-g  ;  if  at  the  third,  it  wants  j^V^'  *"^  ^°  ^"  '  ^®  ^''^^  ^^'® 
can  arrive  as  near  to  unity  as  we  please,  but  can  never  reach  it. 
Unity  then  in  this  case  is  nothing  but  a  limit,  to  which  0,999 

continually  approaches  the  nearer  the  more  figures  it 

has. 

99.  The  preceding  part  of  this  work  contains  all  the  rules 
absolutely  essential  to  the  arithmetic  of  abstract  numbers,  but 
to  apply  them  to  the  uses  of  society  it  is  necessary  to  know  the 
different  kinds  of  units,  which  are  used  to  compare  together, 
or  ascertain  the  value  of  quantities,  under  whatever  form  they 
may  present  themselves.  These  units,  which  are  the  measures 
in  use,  have  varied  with  times  and  places,  and  their  connexion 
has  been  formed  only  by  degrees,  accordingly  as  necessity  and 
the  progress  of  the  arts  and  sciences  have  required  greater 
exactness  in  the  valuation  of  substances,  and  the  construction  of 
instruments, 

TABLES  OF  COIN,  WEIGHT,  AND  MEASURE. 

Denominations  of  Federal  money,  as  determined  by  an  act  of 
Congress,  Aug.  8,  1786f. 

Marked 

10  mills  make  one  cent  c. 

10  cents  one  dime  d. 

10  dimes  one  dollar  5S. 

10  dollars         one  eagle  E. 

t  The  coins  of  federal  money  are  two  of  gold,  four  of  silver, 
and  two  of  copper.  The  gold  coins  are  an  eagle  and  half-eagle ; 
the  silver,  a  dollar^  half-dollar,  double  dime,  and  dime  ;  and  the  cop- 
per a  cent  and  half-cent.  The  standard  for  gold  and  silver  is  eleven 
parts  fine  and  one  part  alloy.  The  weight  of  fine  gold  in  the  eagle  is 
346,268  grains ;  ol  fine  silver  in  the  dollar,  375,64  grains  j  of  copper 
9 


66  Arithmetic, 


English  Money, 

4  farthing  make  1  penny 
12  pence  1  shillin,^ 

20  shillings  1  pound 


£  denotes  pounds. 

s  shillings. 

d  pence. 

q  quarters  or  farthings. 


TROY  WEIGHT. 

24  grains  make        1  penny-weight,  marked  grs.  dwt. 
£0  dwt,  1  ounce,  oz. 

12  oz.  1  pound,  lb. 

By  this  weight  are  weighed  jewels,  gold,  silver,  corn,  bread 
and  liquors. 

apothecaries'  weight. 

20  grains  make     1  scruple,  marked  gr.  sc. 

S  sc.  1  dram  dr.  or  5. 

8  dr.  1  ounce  oz.  or  f . 

12  oz.  1  pound  lb. 

Apotherai'ies  use  this  weight  in  compounding  their  medicines ; 

in  Hy'Oceuts, -^1  lb.  avoirdupois.  The  fine  gold  in  the  half-eagle  ig 
half  the  weight  of  that  in  the  eagle  ;  the  fine  silver  in  the  half-dollar, 
half  the  weight  of  that  in  the  dollar,  &c.  The  denominations  less 
than  a  dollar  are  expressive  of  their  values;  thus,  mj/Hs  an  abbrevia- 
tion of  mi7/e,  a  thousand,  for  1000  mills  are  equal  to  I  dollar;  cent, 
of  centum,  a  hundred,  for  100  cents  are  equal  to  1  dollar;  a  dime  is 
the  French  of  tithe,  the  tenth  part,  for  10  dimes  are  equal  to  1  dollar. 

The  mint-price  of  uncoined  gold,  11  parts  being  fine  and  1  part 
alloy,  is  209  dollars,  7  dimes,  and  7  cents  per  lb,  Troy  weight ;  and 
the  mint-price  of  uncoined  silver,  11  parts  being  fine  and  1  part 
alloy,  is  9  dollars,  9  dimes,  and  2  cents,  per  lb.  Troy, 

In  practical  treatises  on  arithmetic,  may  be  found  rules  for  reducing 
the  Fedeial  Coin,  the  currencies  of  the  several  United  States,  and 
those  of  foreign  countries,  each  to  the  par  of  all  the  others.  It  may 
be  sufficient  here  to  observe  respecting  the  currencies  of  the  several 
states,  that  a  dollar  is  equal  to  63  in  iNew-England  and  Virginia;  8s. 
in  New  York  and  North  Carolina  ;  Ts.  6d  in  New-Jersey,  Pennsyl- 
vania, Delaware,  and  Maryland;  and  48,  8d.  in  Soutb  Carolina  and 
Georgia, 


Tables  of  Coiut  Weight,  and  Measure.  67 

but  they  buy  and  sell  their  drugs  by  Avoirdupois  weight.  Apoth- 
ecaries' is  the  same  as  Troy  weight,  having  only  some  different 
divisions. 

AVOIRDUPOIS  WEIGHT. 

16  drams  make     1  ounce,  marked    dr.  oz. 
16  ounces  1  pound  lb. 

28  lb.  1  quarter  qr, 

4  quarters  1  hundred  weight  cwt. 

20  cwt.  1  ton  T. 

By  this  weight  are  weighed  all  things  of  a  coarse  or  drossy 
nature ;  such  as  butter,  cheese,  flesh,  grocery  wares,  and  all 
metals,  except  gold  and  silver. 

DRY  MEASURE. 

Marked  Marked 


2  pints  make  1  quart  pts.  qts. 

2  quarts  1  pottle         pot. 

2  pottles  1  gallon        gal. 

2  gallons  1  peck  pe. 

4  pecks  1  bushel.       bu. 

2  bushels  1  strike.        str. 


8  bushels     1  quarter  qr. 

5  quarters  1  wey  or  load  wey 
4  bushels  1  coom  or  carnock  co. 
2  cooms  a  seam  or  quarter. 

6  seams    1  wey. 

1|  weys    1  last  L. 


The  diameter  of  a  Winchester  bushel  is  181  Inches,  and  its 
depth  8  inches. — And  one  gallon  by  dry  measure  contains  268| 
cubic  inches. 

By  this  measure,  salt,  lead,  ore,  oysters,  corn,  and  other  dry 
goods  are  measured. 

ALE  AND  BEER  MEASURE. 


Marked  Marked 

2  pints  make    1  quart  pts.  qis. 
4  quarts  1  gallftn        gal. 


8  gallons  1  firkin  oi'  Ale      fir. 

9  gallons  1  firkin  of  Beer    fir. 


2  firkins  1  kilderkin  kil. 

2  kilderkins  1  barrel  bar, 

3  kildeikins  1  hogsliead  hhd. 
3  barrels  1  butt  butt. 


The  ale  gallon  contains  282  cubic  inches.  In  London  the  ale 
firkin  contains  8  gallons,  and  the  beer  firkin  9 ;  other  measures 
being  in  the  same  proportion. 


68 


tArithmetk. 


"WINB  MEASURE. 


Marked 

Q  pints  make  1  quart  pts.  qis. 

4  quarts  1  gallon       gal. 

42  gallons  1  tierce  tier. 
63  gallons  1  hogshead  hhd. 
84  gallons    1  puncheon       pun. 


2  hogsheads  1  pipe  or 

butt 
2  pipes  1  tun 

8  gallons        1  runlet 
1 '  i  gallons     1  barrel. 


Markett 

p.  or  b. 

T. 

run. 

bar. 


By  this  measure,  brandy,  spirits,  perry,  cider,  mead,  vinegar, 
and  oil  are  measured. 

231  cubic  inches  make  a  gallon,  and  10  gallons  make  an  an- 
chor. 


CLOTH  MEASURE. 

Marked  Marke«l 

2|  inches  make  1  nail           nls.|3  qrs.  1  ell  Flemish  Ell  tL 

4  nails                 1  quarter    qrsJs  qrs.  1  ell  English  Ell  Eng. 

4  quarters           1  yard        yds.le  qrs.  1  ell  French.  Ell  Fr. 


lONG  MEASURE. 


3  barley  corns  make  1 


inch 
12  inches 

3  feet 

6  feet 

5i  yards 
40  poles 

8  furlongs 

3  miles 


bar. 
1  foot 
1  yard 
1  fathom 
1  pole 
1  furlong 
1  mile 
1  league 


Marked  Marked 

60  geographical  miles,  or 
c.  in.  69^  statute  miles  1  degree 
ft.  nearly  deg.  or  ^ 

yd.  360  degrees  the  circumfer- 
fath.  ence  of  the  earth, 

pol.  JlsOf  4  inches  make  1  hand, 
fur.    5  feet         1  geometrical  pace, 
mis.    6  points     1  line 
I.  12  lines        1  inch. 


TIME. 


60  seconds  make  1  minute 

s.  or  "  m.  or 
60  minutes     1  hour  h.  or  ° 

24  hours        1  day  d. 

7  days  1  week  w. 


4  weeks         1  month 
13  months,  1  day,  and  6 

hours,  or 
365  days  and  6  hours,  1 

Julian  year. 


y. 


100.  It  is  evident,  that  if  the  several  denominations  of  money, 
weight  and  measure  proceeded  in  a  decimal  ratio,  the  funda- 
mental operations  might  be  performed  upon  these,  as  upon 
abstract  numbers.    This  may  be  shown  by  a  few  examples  in 


deduction. 


69 


Federal  Money.  If  it  were  required  to  find  the  sura  of  !S46,85 
and  !S256,371,  we  should  place  the  numbers  of  the  same  denom- 
ination in  the  same  column,  and  add  them  together  as  in  whole 
numbers;  thus, 

4685 
256371 


303221 
and  the  answer  may  be  read  off  in  either  or  all  of  the  denomina- 
tions; we  may  say  SO  eagles  3  dollars  22  cents  1  mill,  or 
303  dollars  221  thousandths,  or  30322  cents  and  1  tenth,  or 
303221  mills.  It  is  usual  to  consider  the  dollars  as  whole  num- 
bers, and  the  following  denominations  as  decimals.  The  opera- 
tion then  becomes  the  same  as  for  decimals. 


Add    S  34,123 
1,178 

78,001 
61,789 

Sum     gl75,091 


Examples, 


Add    ^456,78 
49,83 
0,22 
7854,394 

Sum     ^8361,224 


From         S542,76 
Subtract      239,481 


Rem. 


303,269 


Multiply  S6,347  by  S4,532. 
Divide  S28,764604  by  ^4,532. 
Divide  S20  by  §2000. 


From         S527,839 
Subtract        22,94 


Rem. 


504,899 


Ms.  §28,764604. 
Ans.  §6,347. 
Ans.  §0,01. 


Reduction. 

101.  When  the  different  denominations  do  not  proceed  in  a 
decimal  ratio,  they  may  all  be  reduced  to  one  denomination,  and 
then  the  fundamental  opei-ations  may  be  performed  upon  this,  as 
upon  an  abstract  number.  If,  fir  example,  the  sum  to  be  oper- 
ated upon  were  £4  15s.  9d,  this  may  easily  be  expressed  in 


7d  Arithmetic, 

pence.  As  1  pound  is  20  shillings,  4  pounds  will  be  4  times  20, 
or  80  shillings.  If  to  this  we  add  the  1  is.  we  shall  have  95s  9d. 
equivalent  to  the  above.  But  as  1  shilling  is  equal  to  12  pence, 
95s.  will  be  equal  to  95  times  12  or  1140  pence.  Adding  9  to 
this,  we  shall  have  1149  pence  as  an  equivalent  expression  for 
£4  15s.  9d.  We  may  now  make  use  of  this  number  as  if  it  had 
no  relation  to  money  or  any  thing  else  ;  and  the  result  obtained 
may  be  converted  again  into  the  different  denominations  by  re- 
versing the  process  above  pursued.  If  it  were  proposed  to  mul- 
tiply this  sum  by  another  number,  sr,  for  instance,  we  should 
find  the  product  of  these  two  numbers  in  the  usual  way  j  thus, 
1149 
37 

8043 
3447 

42513 
42513  is  therefore,  equal  fo  37  times  £4  15s.  9d.  expressed  in 
pence  ;  to  find  the  number  of  pounds  and  shillings  contained  in 
this,  we  first  obtain  the  number  of  shillings  by  dividing  it  by  12, 
which  gives  3542,  and  then  the  number  of  pounds  by  dividing 
this  last  by  20  ;  thus, 


42513 

12 

354,2 
15 

20 

65 

3542 

177 

51 

14     ' 

33 

2 

9 

42513  pence  then  is  equal  to  3542  shillings,  or  to  175  pounds  15 
shillings.  Whence  37  times  £4  1 5s.  9d.  is  equal  to  £177  2s.  9d. 
It  may  be  remarked,  that  shiUings  are  converted  into  pounds  by 
separating  the  right  hand  Jis^ure  and  dividing  those  on  the  left  by  2, 
prefixing  the  remainder,  if  there  be  one,  to  the  figure  separated 
for  the  entire  shiUings,  that  remain.  This  amounts  to  dividing, 
first,  by  10  (90),  and  then  that  quotient  by  2.  If  10  shillings  made 
a  pound  dividing  by  10  would  give  the  number  of  pounds,  but  as 
10  shillings  is  only  half  a  pound,  half  this  number  will  be  the 
number  of  pounds. 


I  Reduction,.  71 

By  a  method  similar  to  that  above  given,  we  reduce  other  de- 
nominations of  money  and  the  different  denominations  of  the 
several  weights  and  measures  to  the  lowest  respectively.  If  it 
were  required  to  find  how  many  grains  there  are  in  2lb.  4oz. 
irdwt,  5grs.  Troy,  we  should  proceed  thus, 

lb.  oz.  dwt.  gta. 

2         4  17  5 

12 

24 
4 

28 
20 

560 
17 

577 
24 

2308 
1154 


13848 
5 


Ans.     13853 
By  dividing  13853  by  24,  and  the  quotient  thence  arising  by 
20,  and  this  second  quotient  by  12,  we  shall  evidently  obtain  the 
number  of  pounds,  ounces,  pennyweights  and  grains  in  13853 
grains.    The  operation  may  be  seen  below. 


13853 

24 

120 

_ 

577 

20 

185 

40   - 

168 

28    12 

177 

24 

173 

160   - 

2 

168 

4 



17 

5 

lb.   oz.  dwt.  gt. 

Result 

2  4  17  5 

72  ^Arithmetic. 

These  examples  will  be  sufficient  to  establish  the  following 
general  rules,  namely ; 

To  reduce  a  compmmd  mimber  to  the  lowest  denomination  con- 
tained in  itf  multiply  the  highest  by  so  many  as  one  of  this  denomi- 
nation makes  of  the  next  lower,  and  to  the  product  add  the  num- 
ber belonging  to  the  next  lower  ;  then  midtiply  this  sum  by  so  many 
as  one  of  this  makes  of  the  third  lower,  adding  the  number  of  the 
loiver,  as  before,  and  so  on  through  the  whole,  and  the  last  sum  will 
be  the  number  required. 

To  reduce  a  number  from  a  lower  denomination  to  a  higher, 
divide  by  so  many  as  it  takes  of  this  lower  denomination  to  make 
one  of  the  higher,  and  the  quotient  will  be  the  number  of  the  higher  ; 
which  may  be  further  reduced  in  the  same  manner  if  there  are  still 
higher  denominations,  and  the  last  quotient  together  with  the  several 
remainders  will  be  equivalent  to  the  number  to  be  reduced. 

Examples  for  practice. 

In  59lb,  13dwt.  5gr.  how  many  grains  ?  dns,  340157. 

In  8012131  grains  how  many  pounds,  &c.  ? 

dns.  1391b.  lloz.  18dwt.  19gr. 

In  1211.  Os.  9|d  how  many  halfpence?  Jus,  58099. 

In  58099  half  pence  how  many  pounds  &r.  ?  Jns.  l'2lL  Os.  9id. 

In  48  guineas  at  28s.  each  how  many  4id.  pence  ? 

Ans.  3584. 

In  one  year  of  S65d.  5h  48'  48"  how  many  seconds  ? 

Ms.  31556928. 

102.  When  we  have  occasion  to  make  use  of  a  number  consist- 
ing of  several  denominations  as  an  abstract  number,  instead  of 
I'educing  tbe  several  parts  to  tlie  lowest  denomination  contain- 
ed in  it,  we  may  reduce  all  the  lower  denominations  to  a  frac- 
tion of  tbe  highest.  Taking  the  sum  before  used,  namely,  4l. 
15s.  9d.  we  reduce  the  lower  denominations,  to  the  higlier,  as 
in  the  last  article  by  division.  Tbe  number  of  pence  9,  or  ^,  is  di- 
vided by  12,  by  multiplying  the  denominator  by  this  number  (54), 
we  have  thus,  -/^s.  which  being  added  to  15s.  or  Y/^-  ^'^^  whole 
number  being  reduced  to  the  form  of  a  fraction  of  the  same 
denominator,  we  have  Y/    ^"d  -j?^,  which  being  added,  make 


Reduction,  "  73 


that  is,  by  multiplying  the  denominator  by  20  (54),  which 
gives  ||§.  >\  hence  £4.  15s.  9d.  is  equal  to  £4i||,  or  £V,V' 
This  may  now  be  used  like  any  other  fraction  and  the  value  of 
the  result  fsiund  in  the  different  denominations.  If  we  multiply 
it  by  37,  we  shall  have  X^ff^S  or  £177-^^^^ ;  and  f^Vir'  reduced 
to  shillings  by  multiplying  the  numerator  by  20,  or  dividing  the 
denominator  by  this  number,  gives  4|s.  or  2  ■^%s.  or  2s.  9d. 

From  the  above  example,  we  may  deduce  the  following  general 
rules,  namely. 

To  reduce  the  several  parts  of  a  compound  number  to  a  fraction 
of  the  highest  denomination  contained  in  it,  make  the  lowest  term 
the  numerator  of  a  fraction,  having  for  its  denominator  the  number 
which  it  takes  of  this  denomination  to  make  one  of  the  next  higher 
and  add  to  this  the  next  term  reduced  to  a  fraction  of  the  same 
denominationf  then  multiply  the  denominator  of  this  sum  by  so  many 
as  make  one  of  the  next  denomination^  and  so  on  through  all  the 
terms,  and  the  last  siim  will  be  the  fraction  requiredj[. 

To  find  the  value  of  a  fraction  of  a  higher  denomination  in  terms  of 
a  lower,  multiply  the  numerator  of  the  fraction  by  so  many  as  make 
one  of  the  lower  denomination,  and  divide  the  product  by  the  denom- 
inator, and  tlie  quotient  will  be  the  entire  number  of  this  denomi- 
nation, the  fractional  part  of  which  may  be  still  further  reduced  in 
the  same  manner. 

To  reduce  2w.  Id.  6h.  to  the  fraction  of  a  month. 

6h.  is  -^-^  of  a  day,  and  being  added  to  one  day,  or  |^d,  gives 
l^d,  the  denominator  of  which  being  multiplied  by  7,  it  becomes 
fW^y.  If  we  now  multiply  the  denominator  of  this  by  4,  we 
shall  have  ■g^p^  of  a  month  as  an  equivalent  expression  for  2w, 
Id.  6h. 

To  find  the  value  of  -f  of  a  mile  in  furlongs,  poles.  Sec, 

t  It  will  often  be  found  more  convenient  to  reduce  the  several 
parts  of  the  compound  number  to  the  lowest  denomination,  as  by  the 
preceding  article,  for  a  numerator,  and  to  take  for  the  denominator 
so  many  of  this  denomination  as  it  takes  to  make  one  of  that,  to 
which  the  expression  is  to  be  reduced ;  thus  41.  15s.  9d.  being  1 149d. 
is  equal  to  i^VVl,  because  Id.  is  |^1. 
10  ' 


74 


Jirithmetie. 

5 

8 

40 

7 

35 

5 

5 

40 

7 

200  - 

14 

28 

60 

56 

21 


dns.  5fur.  28pls.  S^yds. 
Reduce  13s.  6d.  2q.  to  the  fraction  of  a  pound. 


Reduce  6fur.  26pls.  3yds.  2ft.  to  the  fraction  oi  a  mile. 


Reduce  7oz.  4pwt.  to  the  fraction  of  a  pound,  Troy.  Jlns.  |. 
What  part  of  a  mile  is  6fur,  16pls.  ?  *^ns,  |, 

"What  part  of  a  hogshead  is  9  gallons?  w2ns.  ^, 

What  part  of  a  day  is  j^^  of  a  month  ?  Ans.  ^^, 

What  part  of  a  |;enny  is  ^-^  of  a  pound  ?  ^m.  Y< 

Wisat  part  of  acwt.  is  «  of  a  pound,  Avoirdupois  ?  Ans.  -yl^, 
What  part  of  a  pound  is  |  of  a  firthing  ?  dns.  t^\^. 

M'liat  is  the  value  of  f  .,f  a  pound,  Troy  ?     *3ns.  7oz.  4dwt. 
W  hat  is  the  value  of  4  of  pound,  Avoirdupois  ? 

Ans.  9oz.  2^dr 
What  is  the  value  of  |  of  a  cut.  ?  Ms.  3qrs.  3lb.  loz.  124  ir. 
W  hat  is  the  value  of  -^^  ^^  ^  ™i^c  • 

Jlns.  Ifur.  16pls.  2yds.  1ft.  9^',  in. 


What  is  the  value  of  -j?^  of  day  ? 


Jlns,  12h.  55'  23, 


Reduction,  75 

The  several  parts  of  a  compound  number  may  also  be 
reduced  to  the  form  of  a  decimal  fracti<m  of  the  hi.^hest  denomi<- 
nation  contained  in  it,  by  first  finding  tlie  \alue  oi  the  expres- 
sion in  a  vulgar  fraction  as  in  the  lust  article,  and  then  reducing 
this  to  a  decimal,  or  more  conveniently  by  changing  the  terras 
to  be  reduced  into  decimal  parts,  and  dividing  the  numerator 
instead  of  multiplying  the  denominator  by  the  numbers  succes- 
sively employed  in  raising  them  to  the  required  denomination. 

If  we  take  the  sum  already  used,  namely,  £4  15s.  9d.  the 
pence,  9,  maybe  written  |^,  or-^^^*  the  numerator  of  which  ad- 
mits of  being  divided  by  12  witliout  a  remainder.  It  is  thus 
reduced  to  shillings  and  becomes  yVir'^*  ^^  0,75s.  which  added  to 
the  15s.  makes  15,75s.  or  reducing  the  15  to  the  same  denomi- 
nation, Y//'  or  tVAV  »  *"*^  ^^"^  ^^  reduced  to  pounds,  by 
dividing  it  by  20,  the  result  of  which  is  ///^V,  or  0.7875. 
41.  15s.  9d.  therefore  may  be  expressed  in  one  denominatiim, 
thus  4,78751.  and  in  this  state,  it  may  be  used  like  any  other 
number  consisting  of  an  entire  and  fractional  part.  If  it  be 
multiplied  by  37  we  shall  have  for  the  product  177,13751.  This 
decimal  of  a  pound  may  be  reduced  to  shillings  and  pence,  by 
reversing  the  above  process,  or  by  multiplying  successively  by  20 
and  then  by  12. 

0,1375 
20 


9,0000 

The  product  therefore  of  41.  15s,  9d.  by  57  is  1771.  2s.  9d.  as 
before  obtained. 

The  operation,  just  explained,  admits  of  a  more  convenient 
disposition,  as  in  the  following  example. 

To  reduce  19s.  3d.  Sq.  to  the  decimal  of  a  pound. 


4 

12 
20 


3,00 
3,75000 
{9,312500 


0,965625 


7*6  Jtrithmetk. 

Proceeding  as  before,  we  reduce  the  farthings,  3,  considered 
as  4^rl  ***  hundredths  of  a  penny  by  dividing  by  the  figure  on  the 
left,  4,  and  p!a(-e  tiie  quotient,  75,  as  a  decimal  on  tlie  right  of  the 
pence;  \\e  then  take  this  sum,  considered  as  ^^^'L  or  ll^^r^*  ^'^^^ 
is,  annexing  as  many  ciphers  as  may  be  necessary,  and  divide  it  by 
12,  which  brings  it  into  decimals  of  a  shilling.  Lastly,  the  shil- 
lings and  parts  of  a  shilling,  19,3 1 25s.  cotisidered  as  Y^y^YoV^* 
are  reduced  to  decimals  of  a  pound  by  dividing  by  20,  which 
gives  tiie  result  above  found. 

We  may  proceed  in  a  similar  manner  with  other  denomina- 
tions of  money  and  with  those  of  the  several  weights  and  meas- 
ures. One  example  in  these  will  suffice  as  an  illustration  of  the 
method. 

To  reduce  17pls.  1ft.  6in.  to  the  decimal  of  a  mile. 


12 

6 

3 

1,5 

S20 

17,5 

0,00994318  &C. 

The  decimal  in  this,  as  in  many  other  cases,  becomes  period- 
ical (97). 

From  what  has  been  said,  the  following  rules  are  sufficiently 
evident.  To  reduce  a  number  from  a  lower  denomination  to  the 
decimal  of  a  higher,  we  first  change  it,  or  suppose  it  to  he  changed 
into  a  fraction,  having  10,  m'  some  multiple  of  10,  for  its  denomina- 
tor, and  divide  the  numerator  by  so  many  as  make  one  of  this 
higher  denomination,  and  the  quotient  is  the  required  decimal;  which 
together  with  the  whole  number  of  this  denomination,  may  again  be 
converted  into  a  fraction^  having  10  or  a  nmltiple  of  10  for  its  de- 
nominator, and  thus  by  division  be  reduced  to  a  still  higher  name, 
and  so  on. 

Also,  to  reduce  a  decimal  of  a  higher  denomination  to  a  lower, 
tve  multiply  it  by  so  many  as  one  makes  of  this  lower,  and  those 
figures  which  remain  on  the  left  of  the  comma,  when  the  proper 
mimber  arc  separated  for  decimals  (91),  will  constitute  the  whole 
number  of  this  denomination,  the  decimal  part  of  which  may  be  still 
further  reduced,  if  there  he  lower  denominations,  by  multiplying  it 
by  the  number  which  one  makes  of  the  next  denomination,  and  soon. 


Reduction,  77 

It  may  be  proper  to  add  in  this  place,  that  shillings,  pence  and 
farthings  may  readily  be  converted  into  the  fraction  of  a  pound, 
and  the  fraction  of  a  pound  reduced  to  shillings,  pence  and  far- 
things, without  having  recourse  to  the  above  rules.  As  shillings 
are  so  many  twentieths  of  a  pound,  by  dividing  any  given  num- 
ber of  shillings  by  2,  we  convert  them  into  decimals  of  a  pound, 
thus,  15s,  which  may  be  written  Afl.  or  |f^l.  being  divided 
by  2  give  75  hundredths,  or  0,75  of  a  pound.  Also,  as  farthings 
are  so  many  960ths  of  a  pound,  one  pound  being  equal  to  960 
farthings,  the  pence  converted  into  farthings  and  united  with 
those  of  this  denomination,  may  be  written  as  so  many  960ths  of 
a  pound.  If  now  we  increase  the  numerator  and  denominator 
one  twenty  fourth  part,  we  shall  convert  the  denominator  into 
thousandths,  and  the  numerator  will  become  a  decimal. 

Whence,  to  convert  shillings^  pence  and  Jarthings^  into  the  dedmal 
of  a  poundi  divide  the  shillings  by  2,  adding  a  cipher  when  neces- 
sary,  and  let  the  quotient  occupy  the  Jirst  place^  or  first  and  second, 
if  there  he  two  figures,  and  let  the  farthings,  contained  in  the  pence 
and  farthings,  he  considered  as  so  many  thousandths,  increasing  the 
number  by  one,  when  the  number  is  nearer  24  than  0,  and  by  2,  when 
it  is  nearer  48  than  24,  and  so  on. 

Thus,  to  reduce  15s,  9d.  to  the  decimal  of  a  pound,  we  have, 
0,75 
37 

0,787 

This  result,  it  will  be  remarked,  is  not  exactly  the  same  as  that 
obtained  by  the  other  method ;  the  reason  is,  that  we  have  increas- 
ed the  number  of  farthings,  36,  by  only  one,  whereas,  allowing 
one  for  every  24,  we  ought  to  have  increased  it  one  and  a  half. 
Adding  therefore,  a  half,  or  5  units  of  the  next  lower  order,  we 
shall  have  0,78?  5,  as  before. 

On  the  other  hand,  the  decimal  of  a  pound  is  converted  into  the 
lower  denominations,  or  its  value  is  found  in  shillings,  pence,  and 
farthings,  by  doubling  the  first  figure  for  shillings,  increasing  it  by 
one,  when  the  second  figure  is  5,  or  more  than  5,  and  considering 
what  remains  in  the  second  and  third  places,  as  farthings,  after 
having  diminished  them  one  for  every  24. 


7S  Arithmetic. 

In  addition  to  the  rules  that  have  been  given,  it  may  be  observ- 
ed, that  in  those  cases,  where  it  is  required  to  reduce  a  number 
from  one  denomination  to  another,  when  the  two  denominations 
are  not  commensurable,  or  when  one  will  not  exactly  divide  the 
othei ,  it  will  be  found  most  convenient,  as  a  general  rule,  to  re- 
duce the  one,  or  both,  when  it  is  necessary,  to  parts  so  small,  that 
a  certain  number  of  the  one  will  exactly  make  a  unit  of  the 
other.  If  it  were  required,  for  instance,  to  reduce  pounds  to 
dollars,  as  a  pound  does  not  contain  an  exact  number  of  dollars 
without  a  fraction,  we  first  convert  the  pounds  into  shillings,  and 
then,  as  a  certain  number  of  shillings  make  a  dollar,  by  divid- 
ing the  shillings  by  this  number,  we  shall  find  the  number  of 
dollars  required.  A  similar  method  may  be  pursued  in  other 
cases  of  a  like  nature,  as  may  be  seen  in  the  following  examples. 

In  178  guineas  at  28s.  each,  how  many  crowns  at  6s.  8d.  ? 
6s.  8d.  178  5980,8  I     80 

12  28  48       |~^ 

80d.  1424 

356 

4984 
12 

59803 

.^715.  747  crowns  and  4  shillingsf. 

In  this  case,  I  reduce  both  the  guineas  and  the  crown  to  pence, 
and  then  divide  the  former  result  by  the  latter.  In  dividing  by  80, 
I  first  separate  one  figure  on  the  right  of  the  dividend  for  a  deci- 
mal, which  is  the  same  as  dividing  it  by  10,  and  then  divide  the 
figures  on  the  left,  or  the  quotient,  by  8  (47),  joining  what  re- 
mains as  tens  to  the  figure  separated,  to  form  the  entire  remain- 
der, which  is  reduced  back  to  the  original  denomination. 

To  reduce  1 37  five  franc  pieces  to  pounds,  shillings,  &c.  the  franc 
being  valued  at  S0,1796. 

t  Questions  of  this  kind  may  often  be  conveniently  performed  by 
fractions;  thus,  178  guineas,  or  4984s.  divided  by  6s.  8d.  or  6f3.  or 
reducing  the  whole  number  to  the  form  of  a  fraction,  ys.  oecomes 
'*9j-8*  mulMplietl  by  /^  (74),  or  i^^jf 2,  or  ^ *!*'', which  is  equal  to 
747||  J   and  ^|,  or  |,  of  6s.  8d.  is  3  times  }  of  80d.  or  48d.  or  48. 


0,1796 

73,8156 

20 

5 

0,8980 

isr 

36,9078 
20 

18,1560 
12 

6286 
2694 
898 

1,8720 
4 

123,026 
6 

3,4880 

79 


738,156 

Arts.  361.  18s.  id.  3|q.  nearly* 

Examples/or  practice. 

Reduce  7s.  9|d.  to  the  decimal  of  a  pound.         Ms.  0,4025. 

Reduce  3qrs.  2na.  to  the  decimal  of  a  yard.        Ans.  0,875. 

Find  the  value  of  0,852511.  in  shillings,  pence,  &c. 

Jins.  17s.  Od.  2|q.  nearly. 

Reduce  2411.  18s.  9d.  to  federal  money.  Ms.  §967,75, 

Find  the  value  of  0,42857  of  a  month. 

Ms.  Iw.  4d.  23h.  59'  56". 

Required  the  circumf(Prence  of  the  earth  in  English  statute 
miles,  a  degree  being  estimated  at  57008  toisesf. 

Ms.  24855,488. 

We  have  given  rules  for  reducing  a  compound  number  from 
one  denomination  to  another,  as  we  shall  have  frequent  occasion 
in  what  follows  for  making  these  reductions.  They  are  not, 
however  necessary,  except  in  particular  cases,  previously  toper- 
forming  the  fundamental  operations.  The  several  denomina- 
tions of  a  compound  number  may  be  regarded  like  the  different 
orders  of  units  in  a  simple  one,  that  is,  the  number  or  numbers  of 
each  denondnation  may  be  made  the  subject  of  a  distinct  opera- 
tion, the  result  of  which,  being  reduced  when  necessary,  may  be 
united  to  the  next,  and  so  on  through  all  the  denominations. 


t  A  toise  or  French  fathom  is  equal  to  6  French  feet,  and  a  French 
foot  is  equal  to  12,7893  English  inches. 


»0  Jrithnietk, 

Jlddition  of  Compound  J^unnbers. 

103.  The  addition  of  compound  numbers  depends  on  the  same 
principles  as  that  of  simple  numbers,  the  object  being  simply 
to  unite  parts  of  the  same  denomination,  and  when  a  num- 
ber of  these  are  found,  sufficient  to  form  one,  or  more  than  one 
of  a  higher,  these  last  are  retained  to  be  united  to  othei's  of  the 
same  denomination  in  the  given  numbers ;  as  in  simple  addition 
the  tens  are  carried  from  one  column  to  the  next  column  on  the 
left.  We  must,  then,  place  the  compound  numbers,  that  are  to  be 
added,  in  such  a  manner,  that  their  units,  or  parts  of  the  same 
name,  may  stand  binder  each  other  ;  we  must  then  Jind  separately 
the  sum  of  each  column,  always  recollecting  how  many  parts  of 
each  denomination  it  takes  to  make  one  of  the  next  higher.  See  the 
following  example  in  pounds,  shillings  and  pence. 


£ 

s. 

d. 

984 

12 

8 

38 

6 

9 

1413 

14 

10 

319 

18 

2 

2756     12       5 

First,  adding  together  the  pence,  because  they  are  the  parts  of 
the  least  value,  and  taking  together  both  the  units  and  tens  of 
this  denomination,  we  find  29;  but  as  12  pence  make  a  shil- 
ling, this  sum  amounts  to  2  shillings  and  5  pence ;  we  then 
write  down  only  the  5  pence  and  retain  the  shillings  in  order 
to  unite  them  to  the  column  to  which  they  belong. 

Next,  we  add  separately  the  units  and  the  tens  of  the  next  de- 
nomination ;  the  first  give  by  joining  to  them  the  2  sijillirigs  re- 
served from  the  pence,  22  ;  we  write  down  only  the  two  units  and 
retain  the  two  tens  for  the  next  column,  the  sum  of  which,  by  this 
means,  amounts  to  5  tens,  but  as  the  pound,  made  up  of  20  shil- 
lings, contains  2  tens,  we  obtain  the  number  i)f  pounds  result- 
ing from  the  shillings,  by  dividing  the  tens  of  these  last  by  2 ; 
the  quotient  is  2,  and  the  remainder  1,  wliich  last  is  written 
under  the  column  to  which  it  belongs,  while  the  pounds  are  re- 
served for  the  next  column  on  the  left ;  as  this  column  is  the  last. 


Mditicm  of  Compound  JVumherS,  Bi 

the  operation  is  performed  as  in  simple  numbers,  and  the  whole 
sum  is  found  to  be  27561.  12s.  5d. 

The  method  of  proving  the  addition  of  compound  numbers  is 
derived  from  the  same  principles,  as  that  for  simple  numbers, 
and  is  perf(»rmed  in  the  same  manner,  care  being  taken  in  passing 
from  one  denomination  to  another,  to  substitute  instead  of  the 
decimal  ratio,  the  value  of  each  part  in  the  terms  of  that,  which 
follows  it  on  the  right.     Let  there  be,  for  example, 
£      s.       d. 
984     12       8 
38       6       9 
1413     14     10 
319     18       2  . 

2756      H       5 


1122  22  0 
The  operation  on  the  pounds  is  performed  according  to  the 
mile  of  article  19  ;  then  we  change  the  two  pounds  into  tens  of 
shillings,  and  obtain  4  of  these  tens,  which  joined  to  that  written 
under  the  column,  makes  5,  from  which  we  subtract  the  3  units 
of  this  column,  and  place  the  remainder,  2,  underneath,  counting 
it  as  tens  with  regard  to  the  next  column.  There  still  remain 
2  shillings,  which  must  be  reduced  to  pence  ;  adding  the  result, 
24  pence,  to  the  5  that  are  written,  we  have  a  total  of  29,  which 
must  be  again  obtained  by  the  addition  of  all  the  pence,  as  these 
are  the  parts  of  the  lowest  denomination  in  the  question.  This 
really  happens,  and  proves  the  operation  to  be  right. 

Examples. 

£  s.  d.  £  s.  d.  £  s.  d. 

17  13  4  84  17  51  175  10  10 

13  10  2  75  13  4^  107  13  ll| 

10  17  3  51  17  8|  89  18  10 

8  8  7  20  10  lOi  75  12  24 


4  17     15       44  3       3       3 


84 

17 

5^ 

75 

13 

^ 

51 

17 

8| 

20 

10 

104 

17 

15 

44 

10 

10 

11 

261 

5 

H 

24 

23 

20 

8  10      10      11  1  I 


Sum      54        14  261        5       81  452      19 


Proof   23     32     0  24     23     20  232     13 


11 


82 

Arithmetic. 

lb. 

oz.  dwt.  gr. 

lb. 

oz.  dwt. 

gr. 

lb. 

oz.  dwt.  gr. 

17 

3      15      U 

14 

10     13 

20 

27 

10     17      18 

13 

2      13      13 

13 

10      18 

21 

17 

10      IS      13 

15 

3      14      14 

14 

10      10 

10 

13 

11      13        1 

13 

10 

10 

1        2 

3 

10 

1                 2 

12 

1              17 
13     14 

T. 

1 

4       4 
1      19 

4 

T. 

4 

2 

4        3        3 
I 

cwt 

.qr.lb.  oz.  dr. 

,cwt 

qr.lb.  oz, 

.  dr. 

cwt 

.qr.lb.  oz.dr. 

15 

2   15   15    15 

2 

17 

3   13     8 

7 

3 

13 

2   10     7     7 

13 

2   17   13   14 

2 

13 

3   14      8 

8 

2 

14 

1  17     6     6 

12 

2   13    i4   14 

1 

16 

10 

5 

4 

17 

14           6 

10 

1   17  15 

2 

13 

1 

7 

2 

13 

12     7     7 

ls5 

1    10          10 

1 

14 

1      1      2 

2 

3 

13 

10     4     4 

10 

1   12     1     7 

4 

16 

1      7     7 

5 

5 

2   12     8     8 

Mls.fur.pol  yd  ft.  in. 

Mis 

fur  pol.  yd. 

ft.  in. 

Mis  fur.pol.yd.  ft.  in. 

37 

3   14  2   1      5 

28 

2  13  1   ; 

I     4 

28 

3      7  2          7 

28 

4   17  3  2  10 

39 

1    17  2 

2   10 

30 

1         7 

17 

4     4   3   12 

28 

1    14   2 

2 

27 

6  30  2  2 

10 

5     6  3   17 

48 

1    17  2  : 

2     7 

7 

6  20  2   1 

29 

2     2  2          3 

37 

1  29 

3 

5 

2              2   10 

30 

4          2 

2 

20 

2     1 

7  10       2     2 

Subtraction  of  Compound  JVumbers, 

104,  This  operation  is  performed  in  the  same  way  as  the  sub- 
traction of  simple  numbers,  except  with  regard  to  the  number 
which  it  is  necessary  to  borrow  from  the  higher  denominations, 
in   order  to  perform  the  pariial  subtractions,  when  the  lower 
number  exceeds  the  upper.     For  instance, 
£         s.      d. 
from  795       3       0 
take  684     17       4 

Difference     110       5       8 


Subtraction  of  Compound  JWmhers.  83 

In  performing  this  example,  it  is  necesary  to  borrow,  from  the 
column  of  shillings,  1  shilling  or  12  pence,  in  order  to  effect  the 
subtraction  of  the  lower  number,  4,  and  we  have  for  a  remainder 
8  pence.  There  now  remain  in  the  upper  number  of  the  column  of 
shillings  only  2,  it  is  necessary  therefore  to  borrow,  from  that  of 
pounds,  1  pound  or  20  shillings,  we  thus  make  it  22,  of  which, 
when  tlie  lower  number,  17,  is  subtracted,  5  remain  ;  vve  must 
now  proceed  to  the  column  of  pounds,  remembering  to  count  the 
upper  number  less  by  unity,  and  finish  the  opei-ation  as  in  the 
case  of  simple  numbers. 

The  method  of  proving  subtraction  of  compound  numbers,  like 
that  for  simple  numbers,  consists  in  adding  the  difference  to  tlie 
less  of  the  two  numbers. 

Examples  for  practice. 


£      s»  d. 
275  13  4 
176  16  6 

£ 
454 
276 

s.   d. 
14  21 
17  5| 

gr. 

20 
21 

£      «. 

274  14 
85   15 

d. 

7^ 

Rem.  98  16  10 

177 

16  9^ 

188   18 
274   14 

lb.  oz. 

29  3 
20  7 

61 

Proof  275  13  4 

lb.  oz.  dwt.  gr. 
7  3  14  11 
3  7  15  20 

454 

lb. 

27 
20 

14  21 

oz.  dvvt. 

2  10 

3  5 

Ql 

dwt.  gr 

14  5 

15  7 

Rem. 

Proof 

cut.  qr.  lb.  oz.  dr. 

5   17  5  9 
3  3  21  1  7 

cwt.  qr. 

22  2 
20  1 

lb.  oz.  dr. 
13  4  8 

17  6  6 

cwt.  qr.  lb.  or 

21  1  7  6 
13   8  8 

.  dr. 
13 

14 

Rem. 

^ 

Proof 

84  *irithmetic, 

MU.  fur.  pol.  yd.  ft.    in.  Mis.  fur.  pol.  yd.  ft.  in.  Mis.  fur.  pol.  yd  ft.  in. 

14  3  17  1  2   1    70  7  13  1  1  2  70  3  10     7 

10  7  30  2   10    20   14  2  2  7  17  3  11  1  1  3 


Rem. 
Proof 


m.  w.  d.  h.  ' 

m.  w.  d.  h.  ' 

m.  w.  d.  b.  ' 

17  2  5  17  26 

37  1   13  •  1 

71        5 

10     18  18 

15  2   15  14 

17   5  5  f 

Rem. 
Proof 


Multiplication  of  Compound  JVumbers. 

l05.  We  have  seen,  that  a  number  consisting  of  several  denom- 
inations may  be  reduced  to  a  single  one,  either  the  lowest  or  the 
highest  of  those  contained  in  it,  in  which  state  it  admits  of  being 
used  as  an  abstract  number.  But  when  it  is  required  to  find  the 
pi'odurtof  two  numbers,  one  of  which  only  is  compound,  the  sim- 
plest method  is  to  consider  the  multiplication  of  each  denomina- 
tion of  the  compound  number  by  the  simple  factor,  as  a  distinct 
question,  and  the  several  results,  thus  obtained,  will  be  the  total 
product  sought.  If  it  were  proposed,  for  example,  to  multiply 
71.  14s.  7d.  3q.  by  9,  it  may  be  done  thus, 

£  s.  d.  q. 

7  14  7  3 

9  9  9  9 

63  126  63  27 

and  6^\  126s.  63d.  27q.  is  evidently  9  times  the  proposed  sum, 
because  it  is  9  times  each  of  the  parts,  which  compos©  this  sum* 


I  Multiplication  of  Compound  Mimhers.  85 

But  29q.  is  equal  to  6d.  3q.  and  adding  the  6d.  to  the  63d.  we 
have  69d.  equal  to  5s.  9d.  adding  the  5s.  to  the  126s,  we  obtain 
131s,  equal  61.  lis,  and  lastly,  adding  the  61.  to. the  63l.  we  have 
691.  lis.  9d.  3q.  equal  to  the  above  result,  and  equal  to  the  pro- 
duct of 

91.  14s.  7d.  3q.  by  &. 

Instead  of  finding  the  several  products  first,  and  then  reducing 
them,  we  may  make  the  reductions  after  each  multiplication, 
putting  down  what  remains  of  this  denomination,  and  carrying 
forward  the  quotient,  thus  obtained,  to  be  united  to  the  next 
higher  product. 

Hence,  to  multiply  two  numbers  together,  one  of  which  is  com- 
pound, make  the  compound  number  the  multiplicand  and  the  simple 
number  the  midtiplier,  and  beginning  with  the  lowest  denomination 
of  the  muliiplicand,  multiply  it  by  the  multiplier  and  divide  the  pro- 
duct  by  the  number,  which  it  takes  to  make  one  of  the  next  superior 
denomination  ;  putting  down  the  remainder,  add  the  quotient  to  the 
product  of  the  next  denomination  by  the  midtiplier,  reduce  this  sum^ 
putting  down  the  remainder  and  reserving  the  quotient,  as  before, 
and  proceed  in  this  manner  through  all  the  denominations  to  the 
last,  which  is  to  be  multiplied  like  a  simple  number. 

When  the  multiplier  exceeds  12,  that  is,  when  it  is  so  large 
that  it  is  inconvenient  to  multiply  by  the  whole  at  once,  the 
shortest  method  is  to  resolve  it,  if  it  can  be  done,  into  two  or 
more  factors,  and  to  multiply  first  by  one  and  then  that  product 
by  another,  and  so  on,  as  in  the  following  example.  Let  the  two 
numbers  be  £4  13s.  3d.  and  18. 


d. 


13 


41  19  3 

2 


83  18  6 

Here  we  first  find  9  times  the  multiplicand,  or  £41  19s.  3d. 
and  then  take  twice  this  product  which  will  evidently  be  twice 
9^  or  18  times  the  original  multiplicand  (82).    Instead  of  multi- 


ft6  Arithmetic. 

plying  by  9  wc  might  multiply  first  by  3  and  then  that  product 
by  3,  Wijich  would  give  the  same  result ;  also,  the  multiplier  18 
might  be  resolved  into  S  and  6,  which  would  give  the  same  pro- 
duct as  the  above.     If  we  multiply  £S3  18s.  6d.  by  7. 
£  s.         d. 

83  18         6 

>'  7 


587  9  6 

we  shall  have  the  product  of  the  original  multiplicand  by  7  times 
18  or  126. 

If  the  mulHplier  weie  105,  it  miglit  be  resolved  into  7,  3  and  5, 
and  the  product  found  as  above. 

But  it  frequently  hapjiens,  that  the  multiplier  cannot  be  re- 
solve! in  this  way  into  factors.  When  this  is  the  case,  we  may 
take  the  number  nearest  to  it,  which  can  be  so  resolved,  and 
find  the  product  of  the  multiplicand  by  tliis  namber,  as  already 
described,  and  then  add  or  subtract  so  many  times  the  multipli- 
cand, as  this  number  falls  shoi-t,  or  exceeds  the  given  multiplier, 
and  the  result  will  be  the  product  sought.  Let  there  be  £1  7b. 
8d.  to  be  multiplied  by  17. 

£        s.        d. 

17         8 

4 


5 

10          8 
4 

22 
1 

2          8 

7          8 

Product    £23       10         4 

In  the  first  place,  I  find  the  product  of  £1  7s.  8d.  by  16,  which 
is  £22  2s.  8d.  and  to  this  I  add  once  the  multiplicand  and  this 
sura  £?3  10s.  4d.  is  evidently  equal  to  17  times  the  multiplicand. 

106.  It  may  be  observed,  that  in  those  cases,  where  the  de- 
crease of  value  from  one  denomination  to  another,  is  according  to 
the  same  law  throughout,  that  is,  where  it  takes  the  same  number 
of  a  lower  denomination  to  make  one  of  the  next  higher  through 


Multiplication  of  Compound  J^umbers.  87 

all  the  denominations,  the  multiplication  of  one  compound  number 
by  another,  may  be  performed  in  a  manner  similar  to  what  takes 
place  with  regard  to  abstract  numbers. 

This  regular  gradation  is  sometimes  preserved  in  the  denom- 
inations, that  succeed  to  feet  in  long  measure,  1  inch  or  prime 
being  considered  as  equal  to  12  seconds,  and  1  second  to  12  thirds, 
and  so  on,  the  several  denominations  after  feet  being  distinguish- 
ed by  one,  two,  &c.  accents,  thus, 

lOf.    4'    5"    10'". 

If  it  were  required  to  find  the  product  of  2f.  4'  by  3f.  10'  we 
should  proceed  as  below. 

f. 

2  4 

3  10 

1        11  4 

7  0 

8  11  4" 

The  4  inches  or  primes  may  be  considered  with  reference  to 
the  denomination  of  feet,  as  4  twelfths,  or  y\,  and  the  10  inches 
as  -i|,  the  product  of  which  is  -j^,  or  ^|  of  ^\,  or  40",  which 
reduced  gives  3'  4"  ;  putting  down  the  4",  we  reserve  the  3'  to  be 
added  to  the  product  of  2  feet  by  10',  or  i|,  wliich  product  is  || 
of  a  foot,  to  which  3  being  added,  we  have  ||f,  or  If.  and  11' ; 
next  multiplying  4'  or  j\-  by  3,  we  have  4|  or  1,  which  added  to 
the  product  of  2  by  3,  gives  7.  Taking  the  sum  of  these  results, 
we  have  8f.  11'  4",  for  the  product  of  2f.  4'  by  3f.  10'.  The 
method  here  pursued  may  be  extended  to  those  cases,  where  there 
is  a  greater  number  of  denominations. 

AV hence,  to  multiply  one  number  consisting  of  feet,  primes, 
secoiuls,  Sfc.  by  another  of  the  same  kind,  having  placed  the  serceral 
terms  of  the  midtiplier  under  the  corresponding  ones  of  the  midti- 
plicand,  multiply  the  whole  multiplicand  by  the  several  terms  of 
the  midtiplier  successively  according  to  the  rule  of  the  last  article, 
placing  the  first  term  of  each  of  the  partial  products  under  its  res- 
pective multiplier,  and  find  the  sum  of  the  several  columns,  observing 
to  carry  one  for  every  txvelve  in  each  part  qfthe  operation  ;  then  the 


88  •  Arithmetic, 

Jirst  number  on  the  left  will  be  feet^  and  the  second  primes,  and  the 
third  seconds,  and  so  on  regularly  to  tlie  last\. 

Examples  for  practice. 

Multiply  £1  lis.  ed.  2q.  by  5.  Ans,      £7  17s.  8d.  2q. 

Multiply  7s.  4d.  3q.  by  24.  Ans.      £8  17s.  6d. 

Multiply  £1  17s.  6(1.  by  63.  Ans.  £118    2s.  6d. 

Multiply         17s.  9d.  by  47.  Ans.     £41  14s.  3d. 

Multiply  £1    2s.  3d.  by  117.  Ans.  £\  50    3s.  3d. 

What  is  the  value  of  119  yards  of  cloth  at  £2  4s.  Sd.  per 
yard  ?  Ans.  £263  5s.  9d. 

What  is  the  value  of  9cwt.  of  cheese  at  £l  lis.  5d.  per  cwt  2 

Ans.  £14  2s.  9d. 

What  is  the  value  of  96  quarters  of  rye  at  £1  3s.  4d.  per 
quarter.  Ans.  £112. 

What  is  the  weight  of  7  hhds.  of  sugar,  each  weighing  9cwt. 
Sgrs.  12lb.  Ans.  69  cwt. 

In  the  Lunar  circle  of  19  years  of  365d.  5h.  48'  48'"  each,  how 
many  da}'S,  &c.  ?  Ans.  6839d.  14h.  27'  12". 

Multiply  14f.  9'  by  4f.  6'.  Ans.  66f.  4'  6". 

t  The  above  article  relates  to  what  is  coininonlj  called  duodeci- 
mals. The  operation  is  ordinarily  performed  by  beginning  with  the 
highest  denomination  of  the  multiplier,  and  disposing  of  the  several 
products  as  in  the  first  example  below.  The  result  is  evidently  the 
same  whichever  method  is  pursued,  as  may  be  seen  by  comparing 
this  example  with  that  of  the  same  question  on  the  right,  performed 
according  to  the  rule  in  the  text.  This  last  arrangement  seems  to 
be  preferable,  as  it  is  more  strictly  conformable  to  what  takes  place 
in  the  multiplication  of  numbers  accompanied  by  decimals. 

f  /        //  f.  '         " 

10         4         5  10         4         5 

7         8         1  7         8         6 


72         6  11                       s 

6       10  11  4"' 

5  2  2       G"" 

79       11  0  6       6                    79f.  11'       0'^     6'"       6'"' 


5 

2 

2 

6 

6 

10 

11 

4 

72 

G 

11 

Division  of  Compound  J^umhers,  89 


Multiply    4f.  7'  8"  by  9f.  6'.  Ms,  44f.  O'  10". 

Required  the  content  of  a  floor  48f.  6'  long  and  24f.  3'  broad. 

Ans.  1176f.  1'  6". 
What  is  the  number  of  square  feet,  &c.    in  a  marble  slab 
wbose  length  is  5f.  7'  and  breadth  If.  10'  ?       Ans.  lOf.  2'  10". 

Division  of  Compound  Jfumhcrs. 
107.  A  COMPOUND  number  may  be  divided  by  a  simple  num- 
ber, by  regarding  each  of  the  terms  of  the  former,  as  forming  a 
distinct  dividend.  If  we  take  the  product  found  in  article  105, 
namely,  £63  126s.  63d.  27q.  and  divide  it  by  the  multiplier  9, 
we  shall  evidently  come  back  to  the  multiplicand,  £7  14s,  7d.  3q. 
We  arrive  at  the  same  result  also,  by  dividing  tue  above  sum  re- 
duced, or  £69  1  Is.  9d.  3q.  for  we  obtain  one  9th  of  each  of  the  sev- 
eral parts  that  compose  the  number,  the  sum  of  which  must  be  one 
9th  of  the  whole.  But  since,  in  this  case,  each  term  of  the  divi- 
dend is  not  exactly  divisible  by  the  divisor,  instead  of  employing 
a  fraction  we  reduce  what  remains,  and  add  it  to  the  next  lower 
denomination,  and  then  divide  the  sum  thus  formed,  by  the  divi- 
sor.    The  operation  may  be  seen  below. 

£69      lis.     9d.     3q.  i  9 


63  1  £7  14s.  7d.  3q. 


6 
20 


IS] 
9 


41 

36 


69 
63 


27 
27 


12 


90 


Anthmetic. 


Whence,  tu  divide  a  miniher  consisting  of  different  denominations 
by  a  simple  number,  divide  the  highest  term  of  the  compound  num- 
ber by  the  divisor,  reduce  the  re  mainder  to  the  next  lower  denomi- 
nation, adding  to  it  the  number  of  this  denomination,  and  divide  the 
sum  by  the  divisor,  reducing  the  remainder,  as  bfore,  and  proceed  in 
this  way  through  all  the  denominations  to  the  last,  the  remainder 
of  which,  if  there  be  one,  must  have  its  quotient  represented  in  the 
form  of  a  fraction  by  placing  the  divisor  under  it.  The  sum  of  the 
several  quotients,  thus  obtained,  will  be  the  whole  quotient  required. 

When  the  divisor  is  large  and  can  be  resolved  into  two  or 
more  simple  factoi*s,  we  may  divide  first  by  one  of  these  factors 
and  then  that  quotient  by  another,  and  so  on,  and  the  last  quo- 
tient will  be  the  same  as  that  which  would  have  been  obtained 
by  using  the  whole  divisor  in  a  single  operation.  Taking  the 
result  of  the  example  in  the  corresponding  case  of  multiplication, 
^^  e  proceed  thus, 

£83     18s.     6d.     1  2 


o 

£41 

19s. 

3d. 

9 

Q 

36 

o 

2 

£4 

13s. 

Sd, 

— 

5 

1 

20 

20 

- — 

— 

119 

38 

9 

2 

-.- 

29 

18 

27 

18 

— 

__ 

2 

0 

12 

6 

— 

6 

27 

-^ 

27 

By  dividing  £83  18s.  6d.  by  2,  we  obtain  one  haJf  of  this  sum, 
which  being  divided  by  9,  must  give  one  9th  of  one  half,  oi*  one 
18th  of  the  whole.  The  first  operation  may  be  considered  as 
separating  the  dividend  into  two  equal  parts,  and  the  second  a^ 


Division  of  Compound  Mimhers*  91 

distributing  each  of  these  into  nine  equal  parts,  the  number  of 
parts  therefore  will  be  1 8,  and  being  equal,  one  of  them  must  be 
one  18th  of  the  whole. 

But  when  tlie  divisor  cannot  be  thus  resolved,  the  operation 
must  be  performed  by  dividing  by  the  whole  at  once.  If  the 
quotient,  which  we  are  seeking,  were  known,  by  adding  it  to,  or 
subtracting  it  from,  tlie  dividend  a  certain  number  of  times, 
increasing  or  diminishing  the  di\  isor  at  the  same  time  by  as 
many  units,  we  might  change  the  question  into  one,  whose  divi- 
sor would  admit  of  being  resolved  into  factors,  which  would 
give  the  same  quotient ;  we  should  thus  preserve  tlie  analogy 
whicli  exists  between  the  multiplication  and  division  of  compound 
numbers.  But  this  cannot  be  done,  as  it  supposes  that  to  be 
known,  which  is  the  object  of  the  operation. 

Multiplication  and  division,  where  compound  numbers  are 
concerned,  mutually  prove  each  other,  as  in  the  case  of  simple 
numbers.  This  may  be  seen  by  comparing  the  examples,  which 
are  given  at  length  to  illustrate  these  rules. 

Examples  for  practice. 

Divide  £821  irs.  9|d.       by  4.  Ms.  £205  9s.  5id. 

Divide  £28  2s.  l|d.  by  6.  Jlns.  £4  13s.  8id. 

Divide  £57  Ssi  rd.  by  35.  Ans.  £1  12s.  8d. 

Divide  £23  15s.  7|d.  by  37.  Ms.  12s.  lOid. 

Divide  lOGlcwt.  2qrs.        by  28.  Ms.  37c wt.  yqrs.  18lb. 

Divide  375mls.  2fur.  7pls.  2yds.  1ft.  2in.  by  39. 

Jins.  9mls.  4fur.  39pls.  2ft.  Sin. 
If  9  yards  of  clotli  cost  £4  3s.  7id.  what  is  it  per  yard  ? 

Ms.  9s.  3d.  2q. 
If  a  hogshead  of  wine  cost  £33  12s.  what  is  it  per  gallon  ? 

Ms.  10s.  8d. 
If  a  dozen  silver  spoons  weigh  Sib.  2oz.  13pwt.  12grs.  what 
is  the  weight  of  each  spoon. 
If  a  person's  income  be  £150  a  year,  what  is  it  per  day  ? 

Ms.  8s.  2|d.  nearly. 
A  capital  of  £223  16s.  8d.  being  divided  into  96  shares,  what 
is  the  value  of  a  share  ?  Ms.  £2  7s.  S^^^d. 


^  Arithmetic. 

Froportion, 

108.  We  have  shown,  in  the  preceding  part  of  this  work,  the  dit- 
fei'cnt  methods  necessary  for  performing  on  all  numhers,  whether 
whole  or  fractional,  or  consisting  of  different  denominations, 
the  four  fundamental  operations  of  arithmetic,  namely,  addition, 
subtraction,  multiplication  and  division  ;  and  all  questions  rela- 
tive to  numbers  ought  to  be  regarded  as  solved,  when,  by  an 
attentive  examination  of  the  manner  in  wliich  they  are  stated, 
they  can  be  reduced  to  some  one  of  these  operations.  Conse- 
quently, we  might  here  terminate  all  that  is  to  be  said  on  arith- 
metic, for  what  remains  belongs,  properly  speaking,  to  the  prov- 
ince of  algebra.  We  shall,  nevertheless,  for  the  sake  of  exer- 
cising the  learner,  now  resolve  some  questions  which  will  prepare 
him  for  an  algebraic  analysis,  and  make  him  acquainted  with 
a  viry  impor-tant  theory,  that  of  ratios  and  proportions,  which 
is  ordinatily  comprehended  in  arithmetic. 

109.  A  piece  of  cloth  \  3  yards  long  was  sold  for  130  dollars, 
•what  will  he  the  price  of  a  piece  of  the  same  cloth  1 8  yards  long. 

It  is  plain,  that  if  we  knew  the  price  of  one  yard  of  the  cloth 
that  was  sold,  we  might  repeat  this  price  18  times,  and  the 
result  would  be  the  price  of  the  piece  18  yards  long.  Now, 
since  IS  yards  cost  130  dollars,  one  yai*d  must  have  cost  the 
thirteenth  part  of  130  dollars,  or  \y,  performing  the  division, 
we  find  for  the  result  10  dollars,  and  multiplying  this  number  by 
18,  \Ae  have  180  dollars  for  the  answer;  which  is  the  true  cost  of 
the  piece  1 8  yards  long. 

A  couner,  who  travels  always  at  the  same  rate,  having  gone  5 
leagues  in  3  hours,  how  many  wilt  Jie  go  in  11  hours  ? 

Reasoning  as  in  tlie  last  example,  we  see,  that  the  courier 
go<'s  in  one  hour  -I  of  5  leagues,  or  4,  and  consequently,  in  11 
hour's  he  wiil  go  1 1  times  as  much,  or  |  of  a  league  multiplied 
by  11,  or  Y-  that  is  18  leagues  and  1  mile. 

In  how  many  hours  will  the  courier  of  the  preceding  question  go 
22  leagues  ? 

We  see,  that  if  we  knew  the  time  he  would  occupy  in  going  one 
league,  we  should  have  only  to  repeat  this  number  22  times  and 
the  result  would  be  tlie  number  of  hours  required.     Now  the 


Proportion.  93 

courier,  requiring  S  hours  to  go  5  leagues,  will  require  only 
I  of  the  time,  or  |  of  an  liour,  to  go  one  league  ;  this  number, 
multiplied  by  22,  gives  y  or  13  hours  and  |,  that  is,  13  hours 
and  12  minutes. 

110.  We  have  discovered  the  unknown  qu.antities  by  an  anal- 
ysis of  each  of  the  preceding  statements,  but  the  known  numbers 
and  those  required  depend  upon  each  other  in  a  manner,  that  it 
would  be  well  to  examine. 

To  do  this,  let  us  resume  the  first  question,  in  which  it  was 
required  to  find  tlie  price  of  18  yards  of  cloth,  of  which  13  cost 
130  dollars. 

It  is  plain,  that  the  price  of  this  piece  would  be  double,  if  the 
number  of  yards  it  contained  were  double  that  of  the  first ;  that, 
if  the  number  of  yards  were  triple,  the  price  would  be  ti'iple  also, 
and  so  on ;  also  that  for  the  half  or  two  tliirds  of  the  piece  we 
should  have  to  pay  only  one  half  or  two  Uiirds  of  the  whole  price. 

According  to  what  is  here  said,  which  all  those,  wiio  understand 
the  meaning  of  the  terms,  will  readily  admit,  we  see,  that  if  there 
be  two  pieces  of  the  same  cloth,  the  price  of  the  second  ought  to 
contain  that  of  the  first,  as  many  times  as  the  length  of  the 
second  contains  the  length  of  the  first,  and  this  circumstance  is 
stated  in  saying,  that  the  prices  are  in  proportion  to  the  lengths, 
or  have  the  same  relation  to  each  other  as  the  lengths. 

This  example  will  sei've  to  establish  the  meaning  of  several 
terms  which  frequently  occur. 

111.  The  relation  of  the  lengths  is  the  number,  whether  whole 
or  fractional,  which  denotes  how  many  times  one  of  the  lengths 
contains  the  other.  If  the  first  piece  had  4  yards  and  the  second 
8,  the  relation,  or  ratio,  of  the  fijrmer  to  the  latter  would  be  2, 
because  8  contains  4  twice.  In  the  above  example,  the  first  piece 
had  13  yards  and  the  second  18,  the  ratio  of  the  former  to  the 
latter  is  then  i|,  or  l^*^.  In  general,  the  relation  or  ratio  of  two 
numbers,  is  the  quotient  arising  from  dividing  one  bij  the  other. 

As  the  prices  have  the  sauie  relation  to  each  other,  that  the 
lengths  have,  180  divided  by  130  must  give  i|  for  a  quotient, 
which  is  the  case  ;  for  in  reducing  4|^  to  its  most  simple  terror, 
weget4|.  ,^^^  ^  ^^miit\ 


§4  Arithmetic 

The  four  numbers,  13, 18,  130,  180,  written  in  this  order,  are 
then  sucli,  tliat  the  second  contains  the  first  as  many  times  as  the 
fourth  contains  the  third,  and  thus  they  form  what  is  called  a 
proportion. 

AVe  see  also,  that  a  proportion  is  the  combination  of  two  equal 
ratios. 

We  may  observe,  in  this  connexion,  that  a  relation  is  not 
changed  by  multiplying  each  of  its  terms  by  the  same  number ; 
and  this  is  plain,  because  a  relation,  being  nothing  but  the  quo- 
tient of  a  division,  may  always  be  expressed  in  a  fractional  form. 
Tiius  the  relation  \^  is  the  same  as  i|§. 

The  same  considerations  apply  also  to  the  second  example. 
The  courier,  who  went  5  leagues  in  3  hours,  would  go  twice  as 
far  in  double  that  time,  three  times  as  far  in  triple  that  time ; 
thus  11  hours,  the  time  spent  by  the  courier  in  going  18  leagues 
and  |,  or  Y  of  a  league,  ought  to  contain  3  hours,  the  time  re- 
quired in  going  5  leagues,  as  often  as  Y  contains  5. 

The  four  numbers  5,  Y>  3?  11>  ^i'^  then  in  proportion;  andiu 
reality  if  we  divide  Y  ^Y  5,  we  get^l*  a  result  equivalent  to  y. 
It  will  now  be  easy  to  recognise  all  the  cases,  where  there  may 
be  a  proportion  between  the  four  numbers. 

112.  To  denote  that  there  is  a  proportion  between  the  num- 
bers 13, 18,  130  and  180,  they  are  written  thus,  ' 

13  :  18  ::  130  :  180, 
which  is  read  13  is  #o  18  as  130  is  to  180  ;  that  is,  13  is  the  same 
part  of  18  that  130  is  of  180,  or  that  13  is  contained  in  18  as 
many  times  as  130  is  in  180,  or  lastly,  that  the  relation  of  18  to 
13  is  the  same  as  that  of  180  to  130. 

The  first  term  of  a  relation  is  called  the  antecedent,  and  the 
second  the  consequent.  In  a  proportion  there  are  two  antecedents 
and  t  vo  consequents,  viz.  the  antecedent  of  the  first  relation  and 
that  of  the  second  ;  the  consequent  of  the  first  relation  and  that 
of  the  second.  In  the  proportion  13  :  18  :  130  :  180,  the  antece- 
dents are  13,  130  ;  the  consequents  18  and  180. 

We  shall  in  future  take  the  consequent  for  the  numerator,  and 
the  antecedejit  for  the  denominatoi'  of  the  fraction  which  e:^- 
presses  the  relation. 


Proportion,  95 

113.  To  ascertain  that  there  is  a  proportion  between  the  four 
numbers  13,  18,  130  and  180,  we  must  see  if  the  fractions  ^| 
and  m  be  equal,  and,  to  do  this,  we  reduce  the  second  to  its  most 
simple  terms;  but  this  veiification  may  also  be  made  by  con- 
sidering, that  if,  as  is  supposed  by  the  nature  of  proportion,  the 
two  fractions  \^  and  ^^^^  be  equal,  it  follows  that,  by  reducing 
them  to  the  same  denominator,  the  numerator  of  the  one  will  be- 
come equal  to  that  of  the  other,  and  that,  consequently,  18  multi- 
plied by  130  will  give  the  same  product  as  180  by  13.  This  is 
actually  the  case,  and  the  reasoning  by  which  it  is  sho\yn,  being 
independent  of  the  particulai*  values  of  the  numbers,  proves, 
that,  if  four  numbers  be  in  proportion,  tlie  product  of  the  first  ani 
last,  or  of  the  two  extremes,  is  equal  to  the  product  of  the  second  and 
third,  or  of  the  two  means. 

We  see  at  the  same  time,  that,  if  the  four  given  numbers  were 
not  in  proportion,  they  would  not  have  the  abovementioned  pro- 
perty ;  for  the  fraction,  which  expresses  the  first  ratioy  not  being 
equivalent  to  that  which  expresses  the  second,  the  numerator  of 
the  one  will  not  be  equal  to  that  of  the  other,  when  they  are  re- 
duced to  a  common  denominator. 

114.  The  first  consequence,  naturally  drawn  from  what  has 
been  said,  is,  that  the  order  of  the  terms  of  a  proportion  may  be 
changed,  provided  they  be  so  placed,  that  the  product  of  the  ex- 
tremes shall  be  equal  to  that  of  the  means.  In  the  proportion 
13  :  18  :  :  ISO  :  :  180,  the  following  arrangements  may  be  made^ 


13  : 
13: 

;  18  : 
;  130  : 

:  130  : 
:  18  ; 

;  180 
l&O 

180  : 

;  130  : 

:  18  : 

13 

180  : 

;  18  : 

:  130  : 

13 

18  : 

;  13  : 

:  180  : 

130 

130  ; 

;  13  : 

:  180  : 

18 

18  : 

:  180  : 

:  13  : 

130 

130: 

:  180  : 

:  13  : 

18 

fy>r  in  each  one  of  these,  the  product  of  the  extremes  is  fomied  of 
the  same  factors,  and  the  product  of  the  means  of  the  same  fac- 
fDrs.   Th«  second  arrangement,  in  which  the  means  have  chang- 


Arithmelic, 

ed  places  with  each  other,  is  one  of  those  that  most  frequently 
occur*. 

1 15.  This  change  shows  that,  we  may  either  multiply  or  divide 
the  two  antecedents,  or  the  two  consequents,  by  the  same  num- 
ber, without  destroying  the  proportion.  For  this  change,  makes 
the  two  antecedents  to  constitute  the  first  relation,  and  the  two 
consequents,  the  second.  If,  for  instance,  55  :  21  ::  165  :  63, 
changing  the  places  of  the  means  we  should  have, 

55  :  165  :  :  21  :  63  ; 
we  might  now  divide  the  terms,  which  form  the  first  relation,  by 
5,  (111)  which  would  give  11  :  S3  : :  21  :  63,  changing  again  the 
places  of  the  means,  we  should  have  11  :  21  :  :  33  :  63,  a  propor- 
tion which  is  true  in  iiself,  and  which  does  not  differ  from  the 
given  proportion,  except  in  having  had  its  two  antecedents 
divided  by  5. 

116.  Since  the  product  of  the  extremes  is  equal  to  that  of  the 
means,  one  product  may  be  taken  for  the  other,  and,  as  in  divid- 
ing the  product  of  the  extremes,  by  one  extreme,  we  must  neces- 
sarily find  the  other  as  the  quotient,  consequently ^  in  dividing  by 
one  extreme  the  product  of  the  meatis,  we  shall  find  the  other  eX' 
treme.  For  the  same  reason,  if  we  divide  the  product  of  the 
extremes  hy  one  of  the  means,  we  shall  find  the  other  mean. 

*  It  may  be  observed,  that  the  proportion  13  :  ISO  :  :  18  :  180 
might  have  been  at  once  presented  under  this  form,  according  to  the 
solution  of  the  question  in  article  109 ;  for  the  value  of  a  yard  of 
cloth  may  be  ascertained  in  two  ways,  namely,  by  dividing  the  price 
of  the  piece  of  13  yards  by  13,  or  by  dividing  the  price  of  18  yards 
by  18  :  it  follows  then,  that  the  price  of  the  first  must  contain  13  as 
many  times  as  the  price  of  the  second  contains  18  ;  we  shall  then 
have,  IS  :  130  :  :  18  :  180.  We  may  reason  in  the  same  manner 
•with  respect  to  the  2"^  question  in  the  article  above  referred  to,  as 
well  as  with  respect  to  all  others  of  the  like  kind,  and  thence  derive 
proportions  ;  but  the  method  adopted  in  article  J  09  seemed  preferable, 
because  it  leads  us  to  compare  together  numbers  of  the  same  denom- 
ination, whilst  by  the  others  we  compare  prices,  which  are  sums  of 
money,  with  yards,  which  are  measures  of  length ;  and  this  cannot 
be  done  without  reducing  them  both  to  abstract  numbers. 


PYoportion.  97 

We  can  then  find  any  one  term  of  a  proportion,  when  we  know 
the  other  three,  for  the  term  sought  must  be  either  one  of  the 
extremes  or  one  of  the  means. 

The  question  of  article  (109)  may  be  resolved  by  one  of  these 
rules.  Thus,  wlien  we  have  perceived  that  the  prices  of  the 
two  pieces  are  in  the  proportion  of  the  number  of  yards  contain- 
ed in  each,  we  write  the  proportion  in  this  manner, 

13:  18  :  :  130:0?, 
putting  the  letter  x  instead  of  the  required  price  of  18  yards, 
and  we  find  the  price,  which  is  one  of  the  extremes,  by  multiply- 
ing together  the  two  means,  3  8  and  130,  which  makes  2340,  and 
dividing  this  product  by  the  known  extreme,  13  ;  we  obtain,  for 
the  result,  180. 

The  operation,  by  which,  when  any  three  terms  of  a  propor- 
tion are  given,  we  find  the  fourth,  is  called  the  Rule  of  Three. 
Writers  on  arithmetic  have  distinguished  it  into  several  kinds, 
but  this  is  unnecessary,  wlien  the  nature  of  proportion  and 
the  enunciation  of  the  question  are  well  understood  j  as  a  few 
examples  will  sufficiently  show. 

117.  A  person  having  travelled  21 7,5  miles  in  9  days;  it  is 
asked,  how  long  he  will  be  in  travelling  423,9  miles,  he  being 
supposed  to  travel  at  the  same  rate  ? 

In  this  question  the  unknown  quantity  is  the  number  of  days, 
which  ought  to  contain  the  9  days  spent  in  going  217,5  miles, 
as  many  times  as  423,9  contains  217,5  ;  we  thus  get  the  following 
proportion ; 

dajs 

217,5  :  423,9  : :  9  :  x,  and  we  find  for.r,  17,54  nearly. 

118.  All  the  difficulty  in  these  questions,  consists  in  the  man- 
ner of  stating  the  proportion.  The  following  rules  will  be  suffi- 
cient to  guide  the  learner  in  all  cases. 

Among  the  four  numbers  which  constitute  a  proportion,  there 
are  two  of  the  same  kind,  and  two  others  also  of  the  same  kiwd, 
but  different  fi-om  the  first  two.  In  the  preceding  examples,  two 
of  the  terms  are  miles,  and  the  other  two,  days. 

First,  then,  it  is  necessary  to  distinguish  the  two  terms  of 
each  kind,  and  when  this  is  done,  we  shall  necessarily  have  the 
quotient  of  the  greatest  term  of  the  second  kind  by  the  smallest 
13 


m 


98  Jirithmelk. 

of  the  same  kind,  equal  to  the  quotient  of  the  greatest  tei'm  of  the 
first  kind  by  the  smallest  of  the  same  kind,  which  will  give  ue 
this  proportion, 

the  Smaller  term  of  the  first  kind 

is 

to  the  larger  of  the  same  kind 

as 

the  smaller  term  of  the  second  kind 

to  the  larger  of  this  kind. 

In  the  preceding  example  this  rule  immediately  gives, 
217,5  :  423,9  :  :  9  :  a; 
for  the  unknown  term  ought  to  be  greater  than  9,  since  a  greater 
number  of  days  will  be  necessary  to  complete  a  longer  journey, 

119.  If  it  were  required  to  find  how  many  days  it  would  take 
27  men  to  perform  a  piece  of  work,  which  i5  men,  working  at 
the  same  rate,  would  do  in  18  days  ;  we  see  that  the  days  should 
be  less  in  proportion  as  the  number  of  men  is  greater,  and  recip- 
rocally. There  is  still  a  proportion  in  this  case,  but  tlie  order  of 
the  terms  is  inverted ;  for,  if  the  number  of  workmen  in  the 
second  set  were  tri{)lc  of  tliat  in  the  first,  they  would  require 
only  one  third  of  the  time.  Tlie  first  number  of  days  then 
would  contain  the  second,  as  many  times  as  the  second  number 
of  workmen  would  contain  the  first.  This  order  of  the  terms 
being  the  reverse  of  that  assigned  to  them  by  the  enunciation  of 
the  question,  we  say,  that  the  number  of  workmen  is  in  the 
inverse  ratio  of  the  number  of  days.  If  we  compare  the  two  first, 
an?!  the  two  last,  in  tlie  order  in  which  they  present  themselves, 
the  ratio  of  the  former  will  be  3,  or  4»  and  that  of  the  latter  ^, 
which  is  the  same  as  the  preceding  with  the  terms  inverted. 

It  is  evident,  indeed,  that  we  invert  a  ratio  by  inveiting  the 
terms  of  the  fraction,  which  expresses  it,  since  we  make  the  an- 
tecedent take  the  place  of  the  consequent,  and  the  consequent 
that  of  the  antecedent.    |  or  2  :  3  is  the  inverse  of  4,  or  3  :  2. 

The  mode  of  proceeding  in  such  cases,  may  be  rendei-ed  very 
simple ;  for  we  have  only  to  take  the  numbers  denoting  the  two 
sets  of  workmen,  for  the  quantities  of  the  first  kind,  and  the  num- 


Proportion,  99 

hers  denoting  the  days,  for  those  of  the  second,  and  to  place  the 
one  and  the  other  in  the  order  of  their  magnitude ;  proceeding 
thus,  we  have  the  following  proportion, 

15  :27  :  :  X  :  18, 
from  which  we  immediately  find  x  equal  to  10. 

Recapitulating  the  remarks  already  given,  we  have  the  fol- 
lowing rule ;  make  the  number  which  is  of  the  same  kind  with 
the  answer  the  third  term,  and  the  two  remaining  ones  the  jirst 
and  second,  putting  the  greater  or  the  less  first,  according  as  the 
third  is  greater  or  less  than  the  term  sought ;  then  the  fourth  term 
wiU  be  found  by  multiplying  together  the  second  and  third,  and  di- 
viding the  product  by  the  first. 

120.  1st.  A  man  placed  3575  doUai's  at  interest  at  the  rate  of 
5  pr  cent,  yearly  ;  it  is  a.4ked  what  Will  be  the  interest  of  this 
sum  at  the  end  of  one  year  ? 

The  expression,  5  per  cent,  intertftst,  means,  that  for  a  sum  of 
one  hundred  dollars,  5  dollars  is  allowed  at  the  end  of  a  year  ; 
if  then,  we  take  the  two  principals  for  the  quantities  of  the  first 
kind,  and  the  interest  for  those  of  the  second,  we  shall  have, 

100  :  3575  :  :  5  :  X, 
a  i)roportion  which  may  be  reduced  to  20  :  3595  :  :  1  :  x,  ac- 
cording to  the  observation  in  article  115;  then  dividing  the 
two  terms  of  the  first  relation,  by  5,  we  shall  have  4  :  715  :  :  1  :  x, 
whence  x  is  equal  to  ^i«,  or  §178,75  cts. 

We  may  also  resolve  this  question  by  considering  that  5  is  -^-^ 
of  100,  and  that  consequently  we  shall  obtain  the  interest  of  any 
sum  put  out  at  this  rate,  by  taking  the  twentieth  part  of  this  sum. 
Now  ^\  of  S3575  is  gl  78,75  ;  a  result  which  agrees  with  the  one 
before  found. 

2d.  A  merchant  gives  his  note  for  SB00,00  payable  in  a  year; 
the  note  is  sold  to  a  broker,  who  advances  the  money  for  it,  eight 
months  before  the  time  of  payment;  how  much  ought  the  broker 
to  give  ? 

As  the  broker  advances  from  his  own  funds,  a  sum,  which  is 

not  to  be  replaced  till  the  expiration  of  8  months,  it  is  proper 

that  he  should  be  allowed  interest  for  his  money  during  this 

time. 

Let  the  interest  for  a  year  be  6  per  cent,  the  interest  for  8 


100  Arithmetic. 

months  will  be  -^^,  or  |,  of  6,  or  4  ;  a  sum  tlien  of  100  dollars 
lent  for  8  months,  must  be  entitled  to  4  dollars  interest  that  is, 
he  who  borrows  it,  ought  to  return  gl04.  By  considerin.^  the 
SSOO,  as  a  sum  so  returned  for  what  is  advanced  by  the  broker, 
we  shall  have  this  proportion,  104  :  100  :  :  800  :  x,  whence  we 
get  S769,23f  for  the  value  of  a',  that  is,  for  the  sum  the  broker 
ought  to  give.* 

Questions  for  practice. 
What  is  the  value  of  a  cwt.  of  sugar  at  5|d.  per  lb.  ? 

Ms.  21.  1  Is.  4d. 
What  is  the  value  of  a  chaldron  of  coals  at  ll|d.  per  bushel  ? 

Ans  11. 14s.  6d. 
What  is  the  value  of  a  pipe  of  wine  at  lO^d.  per  pint  ? 

Jns,  441.  2s. 
At   31.  9s.  per  cwt.  what  is  the  value  of  a  pack  of  wool, 
weighing  2rwt.  2qrs.  1 3lb.  Ms.  9l.  6d.  J^?^. 

What  is  the  value  of  l|.cwt.  of  coffee  at  5|^d.  per  ounce  ? 

Ms.  611.  12s. 
Bought  3  casks  of  raisins,  each  weighing  2cwt.  2qrs.  25lb, 
what  will  they  come  to  at  2l,  Is.  8d.  per  cwt  ? 

Ms.  171.  43d.  JL2_. 
What  is  the  value  of  2qrs.  Inl.  of  velvet  at  1 9s.  8id.  per 
English  ell  ?  Ms.  8s.  lOld.  i*. 

Bought  12  pockets  of  hops,  each  weighing  Icwt.  2qrs.  17lb.; 
what  do  they  come  to  at  41.  Is.  4d.  per  cwt.  ? 

Ms.  801.  12s.  l|d.  ^Yj!. 
What  is  the  tax  upon  7451.  14s.  8d.  at  3s.  6d.  in  the  pound? 

Ms.  1301.  lOs  Old.  ^W. 

t  A  sum  thus  advanced,  is  called  the  present  worth  of  the  sum  due 
at  the  expiration  of  the  proposed  time. 

*  The  operation  by  which  we  find  what  ought  to  be  given  for  a 
sum  of  money,  when  the  time  of  payment  is  anticipated,  belongs  to 
what  is  called  Discount.  There  are  several  ways  of  calculating 
discount,  but  the  above  is  the  most  exact,  as  it  has  regard  merely  to 
simple  interest. 


Fr(yporiio\u  101 

If  I  of  a  yard  of  velvet  cost,  7s.  3d.  how  many  yards  can  I 
buy  for  Kl.  15s.  6d.  ?  Ms.  28|  yards. 

If  an  ingot  of  gold,  weighing  9lb.  9oz.*12dwt.  be  worth  4111. 
12s.  what  is  that  per  grain  ?  Ms.  l|d. 

How  many  quarters  of  corn  can  I  buy  for  140  dollars  at  4s. 
per  bushel  ?  Ms.  26qrs.  2bu. 

Bouglit  4  bales  of  cloth,  each  cantaining  6  pieces,  and  each 
piece  27  yards,  at  161.  4s.  per  piece ;  what  is  the  value  of  the 
whole,  and  the  rate  per  yard  ? 

Ans.  5881.  16s.  at  12s.  per  yard. 

If  an  ounce  of  silver  be  worth  5s.  6d.  what  is  the  price  of 
a  tankard,  that  weighs  lib.  lOoz.  lOdwt.  4gr.? 

Ms.  61.  3s.  9id.  ^Vtt- 

What  is  the  half  year's  rent  of  547  acres  of  land  at  15s.  6d. 
per  acre  ?  Ms.  2111.  19s.  3d. 

At  §1,75  per  week,  how  many  months  board  can  I  have  for 
lOOl.  ?  Ans.  47m.  2w.  -/J^. 

Bought  1000  Flemish  ells  of  cloth  for  901.  how  must  I  sell  it 
per  ell  in  Boston  to  gain  lOl.  by  the  whole  ?  Ans.  Ss.  4d. 

If  a  gentleman's  income  is  1750  dollars  a  year,  and  he  spends 
19s.  7d.  per  day,  how  much  will  he  have  saved  at  the  year's 
end?  Ans.  I67l.  12s.  Id. 

What  is  the  value  of  1 72  pigs  of  lead,  each  weighing  3cwt. 
2qrs.  ir^lb.  at  81.  17s.  6d.  per  fother  of  19|c\\t.  ? 

Ans.  2861.  4s.  4Ad. 

The  rents  of  a  whole  parish  amount  to  17501.  and  a  tax 
is  granted  of  321.  16s.  6d.  what  is  that  in  the  pound  ? 

If  keeping  of  one  horse  be  ll^d,  per  day,  what  will  be  that 
of  11  horses  for  a  year?  Ans.  1921.  7s.  8id. 

A  person  breaking  owes  in  all  14901.  5s.  lOd.  and  has  in 
money,  goods,  and  recoverable  debts,  7841.  17s.  4d.  if  these 
thitigs  be  delivered  to  his  creditors,  what  will  they  get  on  the 
pound  ?  Ans.  10s.  6id.  ||m. 

What  must  40s.  pay  towards  a  tax,  when  6521.  13s.  4d.  is 
assessed  at  83l.  12s.  4d.  ?  Ans.  5r.  lid.  1||^|. 

Bought  3  tuns  of  oil  for  1511.  14s.  85  gallons  of  which  being 


102  Arithmtlic. 

damaged,  I  desire  to  know  how  I  may  sell  the  rcmaindei*  pei- 
gallon,  so  as  neither  to  gain  nor  lose  by  the  bargain  ? 

Ms.  4s.  6-Vd.  gW 

What  quantity  of  water  must  I  add  to  a  pipe  of  rnouutain 
wine,  valued  at  33l.  to  reduce  the  first  cost  to  4s.  6(1.  per  gallon  ? 

Jlns.  20-|  gallons. 

If  15  ells  of  stuff,  I  yard  wide,  cost  37s.  6d.  what  will  40 
ells  of  the  same  stuff  cost,  being  yar-d  wide  ?     Jim.  61.  1 3s.  4d. 

Shipped  for  Barbadoes  500  pairs  of  stockings  at  3s.  6d.  per 
pair,  and  1650  yards  of  baize  at  Is.  Sd.  per  yard,  and  have 
received  in  return  348  gallons  of  rum  at  6s.  8d,  per  gallon,  and 
750lb.  of  indigo  at  Is.  4d.  per  lb.  what  remains  due  upon  my 
adventure  ?  Ans.  241.  12s.  6d. 

If  100  workmen  can  finish  a  piece  of  work  in  12  days,  how 
many  are  sufficient  to  do  the  same  in  3  days  ?    Ans.  400  men. 

How  many  yards  of  matting,  2ft.  6in.  broad,  will  cover  a 
floor,  that  is  2rft.  long,  and  20ft.  broad  ?  dm.  72  yards. 

How  many  yards  of  cloth,  'Sqrs.  wide,  are  equal  in  meas- 
ure to  30  yards  5qrs.  wide  ?  Jim.  50  yards. 

A  borrowed  of  his  friend  B  2501.  for  7  months,  prosnising 
to  do  him  the  like  kindness  ;  sometime  after  B  had  occasion  for 
3001,  how  long  may  lie  keep  it  to  receive  full  amends  for  tho 
favor  ?  Am.  5  montl»s  and  25  days. 

If,  when  the  pnce  of  a  bushel  ot  wheat  is  6s.  3d.  the  penny 
loaf  weigh  9oz.  what  ought  it  to  weigh  when  wheat  is  at  8s.  2|d, 
per  bushel?  ,4m5.  6oz.  13dr. 

If  4|cwt.  can  be  carried  36  miles  for  35  shillings,  how  many 
pounds  can  be  carried  20  miles  for  the  same  money  ? 

Am.  9071b.  /y. 

How  many  yards  of  canvass,  that  is  ell  wide,  will  line  20 
yards  of  say,  that  is  3qrs.  wide  ?  Am.  12yds. 

If  30  men  can  perform  a  piece  of  work  in  11  days,  how  many 
men  will  accomplish  another  piece  of  work,  4  times  as  big,  in  a 
fiftb  part  of  the  time  ?  Am.  600. 

A  wall  that  is  to  be  built  to  the  height  of  27  feet.  Was  raised  9 
feet  by  12  men  in  6  days  ;  how  many  men  must  be  employed  trt 
finish  the  wall  in  4  days  at  the  game  rate  of  working  ? 

Am.  35. 


Compound  Proportion*  103 

If  40Z.  cost  i|l.  what  will  loz  cost?  *9tis.  ll.  5s.  8(1. 

If  ^\  of  a  ship  cost  2731.  2s.  6(1.  what  is  ■3*2  ^^  l^^**  worth  ? 

Jlns.  2271.  12s.  1(1. 

At  lAl.  per  cwt.  what  does  S^^lb.  come  to?  Jlns-  10|-J. 

If  I  of  a  gallon  cost  |1.  what  will  |-  of  tun  cost  ?     Ans.  1401. 

A  j)erson,  having  |  of  a  coal  mine,  sells  |  of  Ids  share  for 
1711.  what  is  the  whole  mine  worth  ?  Jns.  3801. 

If,  when  the  days  are  1S|  hours  long,  a  traveller  perform  his 
journey  in  35i  days ;  in  how  many  days  will  lie  perform  the 
same  journey,  when  the  days  are  1  lj%  hours  long  ? 

Ans.  40-|-^|  days. 

A  regiment  of  soldiers,  consisting  of  976  men,  are  to  he  new 
clothed,  each  coat  to  contain  2^  yards  of  cloth,  that  is  ifyd. 
wide,  and  to  be  lined  with  shalloon,  |yd.  wide ;  how  many 
yards  of  shalloon  will  line  them  ?        Ans.  4531yds,  Iqr.  2|.nl. 

Compound  Proportion. 

121.  Proportion  is  also  applied  to  questions,  in  which  the  re- 
lation of  the  quantity  recpiired,  to  the  given  quantity  of  the  same 
kind,  depends  upon  several  circumstances,  combined  together ;  it 
is  then  called  Compound  Proportion^  or  Double  Rule  of  Tliree, 
See  some  examples. 

It  is  required  to  find  how  many  leagues  a  person  would  go 
in  17  days,  travelling  10  hours  a  day,  when  he  is  known  to  have 
travelled  112  leagues,  in  29  da}s,  employing  only  7  hours  a  day. 

This  question  may  be  resolved  in  two  ways,  we  will  first  giVe 
the  one  that  leads  to  Compound  Proportion. 

In  each  case,  the  number  of  leagues  passed  over  depends  upon 
two  circumstances,  namely,  the  number  of  days  the  man  travels, 
and  the  number  of  hours  he  travels  in  each  day. 

We  will  not  at  first  consider  this  latter  circumstance,  but  sup- 
pose the  number  of  hours  to  be  the  same  in  each  case ;  the  ques- 
tion then  will  be  ;  a  person  in  29  days^  travels  1 1 2  leagneSf  hoiv 
many  will  he  travel  in  17  days  ?  This  will  furnish  the  follow- 
ing proportion  ; 

29  :  17  :  :  112  :  X, 


104  Aiithmetic. 

The  fourth  term  will  be  equal  to  112  multiplied  by  17  and  divid- 
ed by  29,  or  ^f^*  leagues. 

Now,  to  take  into  consideration  tlie  number  of  hours,  we  must 
say,  if  a  ])erson  travelling  7  hours  a  day,  for  a  certain  number  of 
days,  has  travelled  ^||^'*  leagues,  how  far  will  he  go  in  the  same 
time,  if  he  travel  10  hours  a  day  ?  This  will  lead  to  the  following 
proportion, 

h.        h.  I. 

7:  10  :  i|°4  :  Xy 
which   gives  for  the  fourth  term,  or  answer,   93,793  leagues 
nearly. 

The  question  may  also  be  resolved  by  observing,  that  29  days 
travelling,  at  7  hours  a  day,  is  equal  (o  203  hours  travelling; 
and  that  17  days,  at  10  hours  a  day,  amounts  to  170  hours,*  the 
problem  then  is  reduced  to  this  proportion, 
203  :  170  :  :  112  :  a-, 
by  which  we  find  the  distance  he  ought  to  travel  in  170  hours, 
accojdltig  to  what  he  performed  in  203  hours. 

We  see,  by  the  first  mode  of  resolving  the  question,  that  112 
leagues  has  to  tlie  fourth  term,  or  answer,  the  same  proportion, 
that  29  days  has  to  17,  and  that  7  hours  has  to  10,  stating  this 
in  the  form  of  a  proportion,  we  have 
d.      d. 


a.  a.  •> 

29  :  17  I 

h.        h.    f   • 

7  :  loj 


lea. 
112 


by  which  it  appears,  that  1 12  is  to  be  multiplied  by  both  17  and 
10,  and  to  be  divided  by  both  29  and  7,  that  is  1 12  is  to  be  mul- 
tiplied by  the  product  of  17  by  10,  and  divided  by  the  product 
of  29  by  7,  which  is  the  same  as  the  second  method  of  resolving 
the  question. 

122.  Again,  if  9  labourers,  working  8  hours  a  day,  have 
spent  24  days  in  digging  a  ditch  65  yards  long,  7  wide,  and  5 
deep,  how  many  days  N\ill  it  take  71  labourers  of  equal  ability, 
working  11  huurs  a  day,  to  dig  a  ditch  327  yards  long,  18 
broad  and  7  dcej)  ? 

Here  is  a  question  very  complicated  in  appearance,  but  which 
may  be  resolved  by  proportion. 

If  all  the  conditions  of  these  two  cases  were  alike,  except  the 


Compound  Proportion.  105 

number  of  men  and  the  number  of  days,  the  question  would  con- 
sist only  in  finding  in  how  many  days  71  men  would  perform  the 
work,  which  9  men  have  done  in  24  days ;  we  should  have  then, 

71  :  9  :  :  24  :  x, 
but  instead  of  calculating  the  number  of  days,  we  will  only  indi- 
cate, as  in  article  82,  the  numbers  to  be  multiplied  together,  and 
place  as  a  denominator  those  by  which  they  are  to  be  divided  j  we 
thus  have  for  x  days, 

24  by  9 
71      * 
But  as  the  first  labourers  worked  only  8  hours  a  day  while  the 
others  worked   11,  the  number  of  days  required  by  the  latteu 
will  be  less  in  proportion,  which  will  give 
, ,      „        24  bv  9 

whence  we  conclude  that  the  number  of  days,  in  this  case,  is 
24  bv  9  by  8 
71  by  11    • 
This  number  is  that  of  tlie  days  necessary  for  71  labourers, 
working  11  hours  a  day,  to  dig  the  first  ditch. 

The  ditches  being  of  unequal  length,  as  many  more  days  will 
be  necessary,  as  the  second  is  longer  tlian  the  fii-st,  thus  we  shall 
have 

65:S27::!i^^ii^^:., 
71  by  11  * 

and  the  number  of  days,  this  new  circumstance  being  consider- 
ed, will  be 

24  bv  9  by  8  by  327 
71  by  11  by  65      ' 
Taking  into  consideration  the  breadths,  which  are  not  alike^ 
we  have 

13  •  18  .  .  ^4bv9bv8by327  ^ 

71  by  11  by  65      '  ^' 
and,  consequently,  the  number  of  days  required  changes  td 
24  by  9  by  8  by  327  by  18 
71  by  1 1  by  65  by~T3     ' 
Lastly,  the  depths  being  different,  we  havp, 
14 


106  Arithmetic* 

71  by  1 1  by  05  by  13      '  ^* 
and  the  number  of  days,  resulting  from  the  concurrence  of  all 
the  circumstances,  is 

24  by  9  by  8  by  S£7  by  1 8  by  7 
7 1  by  1 1  by  65  by  13  by  5 
Performing  the  multiplications  and  divisions,  we  get  the  answer 
i-equired,  21  days  m|8|i. 

123.  This  number  is  equal  to  24  multiplied  by  the  fractional 
quantity 

9  bv  8  by  327  by  18  by  7  , 
7.  by  11  by  65  by  13  by  5' 
but  this  last  quantity,  which  expresses  the  relation  of  the  num- 
ber of  days  given,  to  the  number  of  days  required,  is  itself  the 
product  of  the  following  fractions  ; 

9  8         397       187 

TT>  TT»    3-T  »  T7»  T* 

Now,  going  back  to  the  denominations  given  to  these  numbers  in 
the  statement  of  the  question,  we  see  that  these  fractions  are  the 
ratios  of  the  men  and  the  hours,  of  the  lengths,  the  breadths  and 
the  depths,  of  the  two  ditches ;  hence  it  follows,  that  the  ratio  of 
the  number  of  days  given,  to  the  number  of  days  sought,  is  equal 
to  the  product  of  all  the  ratios,  which  result  from  a  comparison 
of  the  terms  relating  to  each  circumstance  of  the  question. 

This  may  be  resolred  in  a  simple  manner  by  first  Snding  the 
value  of  each  of  these  ratios  ;  for,  by  multiplying  together  the 
fractions  that  express  them,  we  form  a  fraction  which  repre- 
sents the  ratio  of  the  quantity  required  to  the  given  quantity  of 
the  same  kind. 

This  fraction,  which  will  be  the  product  of  all  the  ratios  in  the 
question,  will  have  for  its  numerator  the  product  of  all  the  ante- 
cedents, and  for  its  denominator,  that  of  all  the  consequents.  A 
ratio  resulting,  in  this  manner,  from  the  multiplication  of  several 
others,  is  called  a  compound  rath. 

By  means  of  the  fractional  expression 

9  by  8  by  327  by  18  by  7 
71  by  11  by  (id  by  13  by  5* 
and  the  given  number  of  days,  24,  we  make  the  following  propor- 
tion, 


Compound  Proportion,  107 

71  by  11  by  65  by  13  by  5  :  9  by  8  by  327  by  18  by  7  : :  24  :  x, 
whicb  may  also  be  represented  in  this  manner,  as  was  the  preced- 
ing example. 


9^ 


327 
18 


24  :x. 


71 
11 

65 

13 

5 

Our  remarks  may  be  summed  up  in  this  rule  ;  Make  thenumher 
'which  is  of  the  same  kind  with  the  required  answer^  the  third  term  ; 
and  of  the  remaining  numbers,  take  any  two  that  are  of  the  same 
kind  and  place  one  for  a  first  term  and  the  other  for  a  second  term 
according  to  the  directions  in  simple  proportion  ;  tJien  amj  other  two 
of  the  same  kind,  and  so  on,  till  all  are  used  ;  lastly,  multiply  the 
third  term  by  the  product  of  the  second  terms,  and  divide  the  result 
by  the  product  of  the  first  temns,  and  the  quotient  wiU  be  the  fourth 
term,  or  answer  required. 

Examples  for  practice. 

If  lOOl.  in  one  year  gain  51.  interest,  what  will  be  the  interest 
»f  7501.  for  7  years  ?  Ans.  262l.  Is. 

What  principal  will  gain  2621.  10s.  in  7  years,  at  5l.  per  cent, 
per  annum  ?  Jins.  7501. 

If  a  footman  travel  130  miles  in  S  days,  when  the  days  are  12 
hours  long ;  in  how  many  days,  of  10  hours  each,  may  he  travel 
360  miles  ?  ,am.  9||  days. 

If  120  bushels  of  corn  can  serve  14  horses  56  days  ;  how  many 
days  will  94  bushels  serve  6  horses  ?  Ans.  102i|  days. 

If  7oz.  5dwt.  of  bread  be  bought  at  4|d.  when  corn  is  at  4s. 
2d.  per  bushel,  what  weight  of  it  may  be  bought  for  Is.  2d,  when 
the  price  per  bushel  is  5s.  6d.  ?  Ans,  lib.  4oz.  ''||y  'wt. 

If  the  transporation  of  13cwt.  Iqr.  72  miles  be  2i.  10s.  6d. 
what  will  be  the  transportation  of  7cwt.  3qr8.  112  miles  ? 

Ans.  21.  5s.  lid.  lyVVq. 

A  wall,  to  be  built  to  the  height  of  27  feet,  was  raised  to  the 
height  of  9  feet  by  12  men  in  6  days;  how  many  men  must  be 
employed  to  finish  the  wall  in  4  days,  at  the  same  rate  of  work- 
ing ?  *  Ans,  S6  men. 


108  »Snthmeiic, 

If  a  regiment  of  soldiers,  consisting  of  9S9  men  consume  S51 
quarters  of  wheat  in  7  months ;  how  many  soldiers  will  consume 
1464  quarters  in  5  months,  at  that  rate  ?  ^ns.  548S^y^. 

If  248  men,  in  5  days  of  11  hours  each,  dig  a  trench  230 
yards  long,  3  wide  and  2  deep ;  in  how  many  days  of  9  hours 
long,  will  24  men  dig  a  trench  of  420  yards  long,  5  wide  and  3 
deep  ?  Ans.  288^Vy* 

Fellowship^ 

124.  The  object  of  this  rule  is  to  divide  a  number  into  parts, 
which  shall  i)ave  a  given  relation  to  each  other  ;  we  shall  see  in 
the  following  example  its  origin,  and  whence  it  has  is  name. 

Three  merchants  formed  a  company  for  thi^  purpose  of  trade; 
the  first  advanced  25000  dollars,  the  second  18000,  and  the  third 
42000  ;  after  some  time  they  separated,  and  wished  to  divide  the 
joint  profit,  which  amounted  to  57225  dollars  ;  how  much  ought 
each  one  to  have  ? 

To  resolve  this  question  we  must  consider,  tliat  each  man's 
gain  ought  to  have  the  same  relation  to  the  whole  gain,  as  the 
money  he  advanced  has  to  the  whole  sum  advanc  ed  ;  for  he,  who 
furnishes  a  half  or  third  of  this  sum,  ought,  plainly,  to  have  a 
half  or  third  of  the  profit.  In  the  present  example,  the  whole 
sum  being  85000  dollars,  the  particular  su-ns  will  be  respec- 
tively |4-n^  mn  4lo-^^  «f  it;  and  if  vne  multiply  these 
fractions  by  the  whole  gain,  57225,  we  shall  obtain  the  gain  be- 
longing to  each  man.  It  is  moreover  evident,  that  the  sum  of 
the  parts  will  be  equal  to  the  whole  gain,  because  ihfi  sum  of  the 
above  fractions,  having  its  numerator  equal  to  its  denominator, 
is  necessarily  an  unit. 

We  have  therefore,  these  proportions ; 

85000  :  25000  :  :  57225  :  to  the  first  man's  gain, 
85000  :  18000  :  ;  57225  :  to  the  second  man's  gain, 
85000  :  42000  :  :  57225  :  to  the  third  man's  gain, 

which  may  he  enunciated  thus  ; 

The  whole  sura  advanced  :  to  each  mat.'s  particular  sum  :  :  the 

whole  gain  :  to  each  man's  particular  gain. 


i  Fellowship.  109 

By  simplifying  the  first  ratio  of  each  of  these  proportions  we 
have 


85  :  25  :  :  57225  :  to  the  gain  of  the  P*-  or  gl6830||, 
85  :  18  :  :  57225,:  to  the  gain  of  the  2'^-  or  Sl2118||, 
85:42::  57225  :  to  the  gain  of  the  3''-  or  S28275|.|. 
If  all  the  sums  advanced  had  been  equal,  the  operation  would 
have  been  reduced  to  dividing  the  whole  gain  by  the  number  of 
sums  advanced ;   we  may   reduce  tlie  question  to  this  in  the 
present  case,  by  supposing  the  whole  sum,  §85000,  divided  into 
85  partial  sums,  or  stocks  of  §1000  each,  the  gain  of  each  of 
these  sums  will  evidently  he  the  85*-  part  of  the  whole  gain ;  and 
nothing  remains  to  be  done,  except  to  multiply  this  part  severally 
by  25,  18,  and  42,  considering  the  sums  25000,  1 8000  and  42060 
as  the  amounts  of  25  shares,  18  shares  and  42  shares- 

In  commercial  language  the  money  advanced  is  called  the 
capital  or  stock,  and  the  gain  to  be  divided,  the  dividend. 
The  following  question  is  very  similar  to  that  just  resolved, 

125.  It  is  required  to  divide  an  estate  of  67250  dollars  among 
3  heirs,  in  such  a  manner,  that  the  share  of  the  second  shall  be 
I  of  that  of  the  first,  and  the  share  of  the  third  |  of  that  of  the 
second. 

It  is  plain  that  the  share  of  the  third,  compared  with  that  of  the 
first,  will  be  I  of  |  of  it,  or  /„  ;  tlien  the  three  parts  required 
will  be  to  each  other  in  the  proportion  of  the  numbers  1,  |  and 
■J-^,  Reduciner  these  to  a  common  denominator,  we  find  them 
|o,  ^8^,  and  j\,  and  have  the  three  numbers  20,  8  and  7,  which 
are  proportional  to  the  first ;  but  as  their  sum  is  35,  it  is  plain, 
that  if  we  take  three  parts  expressed  by  the  fractions,  |^,  -j^, 
and  -j'-y,  they  will  be  in  the  required  proportion.  The  question 
will  then  be  resolved  by  taking  ||,  then  -^^  and  then  -^\  of  67250 
dollars,  which  will  give  the  sums  due  to  the  heirs,  according  to 
the  manner  prescribed,  namely ; 

g38428||,  §l537I-i|,  and  §13450. 

126.  Again,  there  are  two  fountains,  the  first  of  which  will 
fill  a  certain  reservoir  in  2i  hours,  and  the  second  will  fill  the 
pame  reservoir  in  3|  hours  ;  how  much  time  will  be  rccpured  to 


110  drithmetic. 

fill  the  reservoir,  by  means  of  both  fountains  running  at  the  same 
time  ? 

We  must  first  ascertain  what  part  of  the  reservoir  will  be  filled 
by  the  first  fountain  in  any  given  time,  an  hour  for  instance.  It 
is  evident  that,  if  we  take  the  contents  of  the  reservoir  for  unity, 
we  have  only  to  divide  1  by  2i,  or  |,  which  gives  us  |  for  the 
part  filled  in  one  hour  by  the  first  fountain.  In  the  same  man- 
ner, dividing  1  by  S|,  or  y,  we  obtain  ■*-^  for  the  part  of  the 
reservoir  filled  in  an  hour  by  the  second  fountain ;  consequently, 
the  two  fountains,  flowing  together  for  an  hour,  will  fill  |  added 
to  YJ1  or  ^^  of  the  reservoir.  If  we  now  divide  1,  or  the  con- 
tents of  the  reservoir,  by  ||,  we  shall  obtain  the  number  of 
hours  necessary  for  filling  it  at  this  rate ;  and  shall  find  it  to  be 
^1^,  or  an  hour  and  a  half. 

Authors  who  have  written  upon  arithmetic,  have  multiplied  and 
varied  these  questions  in  many  ways,  and  have  reduced  to  rules 
the  processes  which  serve  to  resolve  them  ;  but  this  multiplica- 
tion of  precepts,  is,  at  least,  useless,  because  a  question  of  the 
kind  we  have  been  considering,  may  always  be  solved  with  facil- 
ity by  one  who  perceives  what  follows  from  the  enunciation  ; 
especially  wlien  he  can  avail  himself  of  the  aid  of  algebra ;  we 
shall  therefore  proceed  to  another  subject. 

Besides  the  proportions  composed  of  four  numbers,  one  of  the 
two  first  of  which  contains  the  other  as  many  times  as  the  cor- 
responding one  of  the  two  last,  contains  the  other;  it  has  been 
usual  to  consider  as  such  the  assemblage  of  four  numbers,  such 
as  2,  7,  9,  14,  of  which  one  of  the  two  first  exceeds  the  other  as 
much  as  the  corresponding  one  of  the  two  last,  exceeds  the  other. 

These  numbers,  which  may  be  called  equidifferenU  possess  a 
remarkable  property,  analogous  to  that  of  proportion,  for  the  sum 
of  the  extreme  terms,  2  and  14,  is  equal  to  the  sum  of  the  means, 
7  and  9*. 

*  The  ancients  kept  the  theory  of  proportions  very  distinct  from 
the  operations  of  arithmetic.  Euclid  gives  this  theory  in  the  fifth 
book  of  his  elements,  and  as  he  applies  the  proportions  to  lines,  it  is 
apparent,  that  we  thence  derive  the  name  of  geometrical  proportion  ; 


I  Fellowship.  Ill 

To  show  this  property  to  be  general,  we  must  observe,  that  the 
second  term  is  equal  to  the  first  increased  by  the  difference,  and 
that  the  fourth  is  equal  to  the  third  increased  by  the  difference ; 
hence  it  follows,  that  the  sum  of  the  extremes,  composed  of  the 
first  and  fourth  terms,  must  be  equal  to  the  first  increased  by  the 
third  increased  by  the  difference.  Also,  that  the  sum  of  the 
means,  composed  of  the  second  and  third  terms,  must  be  equal 
to  the  first  increased  by  the  difference  increased  by  the  third 
term ;  these  two  sums,  being  composed  of  the  same  parts,  must 
consequently  be  equal. 

We  have  gone  on  the  supposition,  that  the  second  and  fourth 
terms  were  greater  than  the  first  and  tliird  ;  but  the  con- 
trary may  be  the  case,  as  in  the  four  numbers  8,  5,  15,  12  ;  the 
second  term  will  be  equal  to  the  first  decreased  by  the  difference, 
and  the  fourth  will  be  equal  to  the  third  decreased  by  the  differ- 
ence. By  changing  the  word  increased  into  decreased,  in  the 
preceding  reasoning,  it  will  be  proved  that,  in  the  present 
case,  the  sum  of  the  extremes  is  equal  to  that  of  the  means. 

We  shall  not  pursue  this  theory  of  equidifferent  numbers  fur- 
ther, because,  at  present,  it  can  be  no  use. 

Questions  for  practice, 
A  and  B  have  gained  by  trading  §182.     A  put  into  stock 
jgSOO  and  B  S400 ;  what  is  each  person's  share  of  the  profit  ? 

Ans,  A  S78  and  B  gl04. 

and  that  tne  aaine  of  arithmetical  proportion  was  given  to  an  assem- 
blage of  equidifferent  numbers,  which  were  not  treated  of  till  a  much 
later  period.  These  names  are  very  exceptionable ;  the  word  propor- 
tion has  a  determinate  meaniag,  which  is  not  at  all  applicable  to 
equidifferent  numbers.  Besides,  the  proportion  called  geometrical,  is 
not  less  arithmetical  than  that  which  exclusively  possesses  the  latter 
name.  M.  Lagrange,  in  his  Lectures  at  the  Normal  school,  has  pro- 
posed a  more  correct  phraseology,  and  I  have  thought  proper  to 
follow  his  example. 

Equidifference^  or  the  assemblage  of  four  equidifferent  numbers,  or 
arithmetical  proportion,  is  written  thus  ;  2.7:9.14. 

Among  English  writes  the  following  form  is  used  j 
S  .  .  r  : :  9  .  .  14. 


11£  Arithmetic, 

Divide  gl20  between  three  persons,  so  that  their  shares  shall 
jbe  to  each  other  as  1,  2,  and  3,  respectively. 

Ans.  $5.0,  ^40,  and  jg60. 

Three  persons  make  a  joint  stock.  A  put  in  Si 85,66,  B 
g98,50,  and  C  !g*6,85  ;  they  trade  and  gain  S222  j  what  is  each 
person's  share  of  the  gain  ? 

Ans.  A  Sl04,177/-rVT' »  S60,57-5WVt»  a^^  C  47,25|||^f 

Three  merchants  A,  B,  and  C,  freight  a  ship  with  340  tuns  of 
wine  5  A  loaded  110  tuns,  B  97,  and  C  the  rest.  In  a  storm  the 
seamen  were  obliged  to  throw  85  tuns  overboard ;  how  much 
must  each  sustain  of  the  loss  ?     Ms.  A  27|,  B  241,  and  C  33^. 

A  ship  worth  S860  being  entirely  lost,  of  which  |  belonged  to  A, 
^  to  B,  and  the  rest  to  C  j  w  hat  loss  will  each  sustain,  supposing 
S500  of  her  to  be  insured  ?     Ans.  A  S45,  B  S90,  and  C  S225. 

A  bankrupt  is  indebted  to  A  8277,33  ,  to  B  ^305,17,  to  C 
gl52,  and  to  D  glOS.     His  estate  is  worth  only  ^677,50;  how 
must  it  be  divided  2         Am.  AS223,8 1 1|  ||,  B  S246,28^^Vt> 
C  Sl22,6P|m,  and  D  884,73||||. 

A  and  B,  venturing  equal  sums  of  money,  clear  by  joint  trade 
^154.  By  agreement  A  vi^as  to  have  8  per  cent,  because  he 
spent  his  time  in  the  execution  of  the  project,  and  B  was  to  have 
only  5  per  cent. ;  what  was  A  allowed  for  his  trouble  ? 

Ans»  835,534^. 

Three  graziers  hired  a  piece  of  land  for  S60,50.  A  put  in  5 
sheep  for  4|  months,  B  put  in  8  for  5  months,  and  C  put  in  9  for 
6|  months ;  how  much  must  each  pay  of  the  rent  ? 

Am.  A  §11 ,25  ,  B  S20,  and  C  §29,25. 

Two  merchants  enter  into  partnership  for  18  months;  A  put 
into  stock  at  first  §200,  and  at  the  end  of  8  months  he  put  in 
§100  more ;  B  put  in  at  first  §550 ,  and  at  the  end  of  4  months 
took  out  §140.  Now  at  the  expiration  of  the  time  they  find  they 
have  gained  §526  ;  what  is  each  man's  just  share  ? 

Am.  A's  §192,95^|«:5. 
B's     333,041||*, 

A,  with  a  capital  of  §1000,  began  trade  January  1,  1776,  and 
meeting  with  success  in  business  he  took  in  B  a  partner,  \^  ith  a 
capital  of  §1500  on  the  first  of  March  following.    Three  months 


Alligation,  11$ 

after  that,  they  admit  C  as  a  third  partner,  who  brought  into 
stock  S2800,  and  after  trading  together  till  the  first  of  the  next 
year,  they  find  the  gain,  since  A  comtnenced  husiness,  to  be 
Sl776,50.     How  must  this  be  divided  among  the  partners  ? 

Ans.  A's  S457,4f,|8*. 
B's  571,832  2  2. 
C'8     747,19|^|. 


128.  We  shall  not  omit  the  rule  of  alligation,  the  object  of 
which  is  to  find  the  mean  value  of  several  things  of  the  sane 
kind,  of  different  values ;  the  following  examples  will  sufficiently 
illustrate  it. 

A  wine  merchant  bought  several  kinds  of  wine,  namely ; 
^  ISO  bottles  which  cost  him  10  cents  each, 

^^*  75  at  15 

231  at  12 

27  at  20; 

lie  afterwards  mixed  them  together ;  it  is  required  to  ascertain 
the  cost  of  a  bottle  of  the  mixture.  It  will  be  easily  perceived, 
that  we  have  only  to  find  the  whole  cost  of  the  mixture,  and  the 
number  of  bottles  it  contains,  and  then  to  divide  the  first  sum, 
by  the  second,  to  obtain  the  price  required. 

Now,  the  130  bottles  at  10  cents  cost     ISOO  cent« 
75  at  15  cost     1125, 

231  at  12  cost     2772, 

27  at  20  cost       540, 


tbeu  463  bottles  cost  5737  cents, 

3737  divided  by  463  give  12,39  cents,  the  price  of  a  bottle  of 

the  mixture. 

The  preceding  rule  is  also  used  for  finding  a  mean  of  differ- 
ent results,  given  by  experiment  or  observation,  which  do  not 
agree  with  each  other.  If,  for  instance,  it  were  required  to 
know  the  distance  between  two  points  considerably  removed, 
from  each  other,  and  it  had  been  measured ;  whatever  care 
might  have  been  used  in  doing  this,  tjiera  w»h1(1,  always  be  a 
15 


114  Arithmetic. 

little  uncertainty  in  the  result,  on  account  of  the  errors  inev- 
itably committed  by  the  manner  of  placing  the  measures  one 
after  the  other. 

We  will  suppose  that  the  operation  has  been  repeated  several 
times,  in  order  to  obtain  the  distance  exactly,  and  that  twice  it 
has  been  found  3794yds,  2ft.  that  three  other  measurements  have 
given  3795yds.  1ft.  and  a  thiid  3793yds.  As  these  numbers  are 
not  alike,  it  is  evident  th^it  some  must  be  wrong  and  perhaps  all. 
To  obtain  the  means  of  diminishing  the  error,  we  reason  thus  ; 
if  the  true  distance  had  been  obtained  by  each  measurement,  the 
sum  of  the  results  would  be  equal  to  six  times  that  distance,  and 
it  is  plain  that  this  would  also  be  the  case,  if  some  of  the  residts 
obtained  were  too  little,  and  others  too  great,  so  that  the  increase, 
produced  by  the  addition  of  the  excesses,  should  make  up  for 
what  the  less  measurements  wanted  of  the  true  value.  We 
should  then,  in  this  last  case,  obtain  the  true  value  by  dividing 
the  sum  of  the  results  by  the  number  of  them. 

This  case  is  too  pei  uliar  to  occur  frequently,  but  it  almost 
always  happens,  that  the  errors  on  one  side  destroy  a  part  of 
those  on  the  other,  and  the  remainder,  being  equally  divided 
among  the  results,  becomes  smaller  according  as  the  number  of 
results  is  greater. 

According  to  these  considerations  we  shall  proceed,  as  follows  ; 

yds.       ft.  a 

We  take  twice  37.4  2        or        7589  1 

yds.      ft, 

3  times  3795  i         or       11386  0 

yds. 

once  3793  or  5793 


6  results,  giving  in  all   227o8  1. 

Dividing  22768yds.  1ft.  by  6,  we  find  the  mean  value  of  the 
re«[uired  distance  to  be  S794yds.  2ft. 

129.  Questions  somctiiries  occur,  which  are  to  be  solved  by  a 
method,  the  reverse  of  that  above  given.  It  may  be  required, 
for  example,  to  find  what  quantity  of  two  diflcrent  ingicdients  it 
w  >uld  take  to  make  a  mixture  of  a  certain  value.  It  is  evident, 
that  if  the  value  of  the  mixture  required  exceeds  that  of  one  of 
the  ingredients  just  as  much  as  it  falls  short  of  that  of  the  other, 
we  should  take  equal  quantities  of  each  to  make  the  compound* 


Migntiou.  ItS 

So  also,  if  the  value  oftlie  mixture  exceeded  that  of  one  twice  as 
murh  as  it  fell  short  of  that  of  the  otiier,  the  proportion  of  the 
ingredients  would  be  as  one  half  to  one.  To  mix  wine  at  $2  per 
gallon  with  wine  at  S3,  so  that  the  compound  shall  be  worth 
^3,50,  we  should  take  equal  quantities,  or  quantities  in  the 
proportion  of  1  to  1.  If  the  mixture  were  required  to  be  worth 
g2,66|,  the  quantities  would  be  in  the  proportion  of  |  to  1,  or  of 

-r-j-  to  ■;:^Y  >  ^^^  generally,  the  nearer  the  mixture  rate  is  to 

that  of  one  of  the  ingredients,  the  greater  must  be  the  quantity  of 
this  ingredient  with  respect  ti)  the  other,  and  the  rev  erse  ;  hence, 
Tojind  the  p-oportion  of  two  ingredients  oj  a  given  valuer  neces- 
sary to  constitute  a  compoiind  of  a  required  valuer  make  the  differ- 
ence between  the  value  of  each  ingredient  and  that  of  the  compound 
the  denominator  of  a  fraction^  whose  numerator  is  one,  and  these 
fractions  will  express  the  proportion  required  ;  and  being  reduced 
to  a  common  denominator,  the  numerators  will  express  the  same 
proportion,  or  show  what  quantity  of  each  ingredient  is  to  be 
taken  to  make  t!ie  required  compound. 

When  the  compound  is  limited  to  a  certain  quantity,  the  pro- 
portion of  the  ingredients,  corresponding  to  it,  may  be  found  by 
«aying;  as  the  whole  quantity,  found  us  above,  is  to  the  quantity 
required,  so  is  each  part,  as  obtained  by  the  rule,  to  the  required 
quantity  of  each. 

Let  it  be  required,  for  example,  to  mix  wine  at  5s.  per  gallon 
and  83.  per  gallon,  in  such  quantities  that  there  may  be  60  gal- 
lons worth  6s.  per  gallon.  The  difference  between  6s.  and  5s. 
is  1,  and  between  6s.  and  8s.  is  2,  giving  for  the  required  quan- 
tities the  ratio  of  |  to  |,  or  2  to  I ;  thus,  taking  x  equal  to  the 
quantity  at  5s.  and  1/ equal  to  the  quantity  at  8s.  we  have  these 
proportions ;  3  :  60  :  :  2  :  x,  and  S  :  60  :  :  1  :  t/,  giving,  for  the 
answer,  40  gallons  at  5s.  and  20  gallons  at  8s.  per  gallon. 

Also,  when  one  of  the  ingredients  is  limited,  we  may  say;  as 
the  quantity  of  the  ingredient  found  as  above,  is  to  the  required 
quantity  of  the  same,  so  is  the  quantity  of  the  other  ingredient 
to  the  proportional  part  required. 

For  example,  I  would  know  how  many  gallons  of  water  at 
Os,  per  gallon,  I  must  mix  with  thirty  gallons  of  wine  at  6s.  per 


116  Arithmetic. 

gallon,  so  that  the  compound  maybe  worth  5s.  per  gallon.  Firet, 
the  difference  between  Os.  and  5s.  is  5  :  and  the  difference  be- 
tween 6s.  and  5s.  is  1  :  the  quantity  of  water  therefore  will  be  to 
that  of  the  wine,  as  ^  to  \f  or  as  1  to  5.  Then,  from  this  ratio, 
we  institute  the  proportion,  5  :  30  :  :  1  :  x,  which  gives  6,  for 
the  number  of  gallons  required. 

As  we  have  found  the  proportion  of  two  ingredients  necessary 
to  form  a  compound  of  a  required  value,  so  also  we  may  con- 
sider either  of  these  in  connexion  with  a  third,  with  a  fourth, 
and  so  on,  thus  makitig  a  compound  of  any  required  value,  con- 
sisting of  any  number  whatever  of  simple  ingredients.  The  two 
ingredients  used,  however,  must  always  be,  one  of  a  greater  and 
the  other  of  a  less  value,  than  that  of  the  compound  required. 

A  grocer  would  mix  teas  at  12s.  and  10s.  with  40lbs.  at  4s.  per 
pound,  in  such  proportions  that  the  composition  shall  he  worth  8s» 
per  Ih.  If  he  mix  only  two  kinds,  the  one  at  4s.  and  the  one  at 
10s.  their  quantities  will  be  in  the  ratio  of  1  to  i.  or  1  :  2  ;  and 
if  he  mix  the  tea  at  4s.  also  with  that  at  12s.  their  ratio  will  be 
that  <»f  I  to  1,  or  of  1  to  1.  A<lding  together  the  proportions  of 
the  ingredient,  which  is  taken  with  each  of  the  others,  we  find 
the  several  quantities,  at  4s.  10s.  and  12s.  t)  be  as  2,  2,  and  1.. 
And  takirg  x  for  the  number  of  lbs,  at  10s.  and  y  for  the  quantity 
at  lis.  we  have  the  following  proportions; 

2  :  40  :  :  2  :  .T  ;  and  2  :  40  :  :  1  :  i/ ; 
giving,  for  the  answer,  40lb.  at  lOs  and  20lb.  at  I2s.  per  pound. 

The  problems  of  the  two  last  articles  are  generally  distin- 
guished by  the  names  of  alligation  medial,  and  alligation  alter- 
nate. A  full  explanation  of  the  latter  belongs  properly  to  algebra. 

Examples. 

A  composition  being  made  of  5lb.  of  tea  at  7s.  per  pound,  9lb. 
at  8s.  6d.  per  pound,  and  14ilb.  .at  5s.  lOd.  per  pouiid  j  what  is 
a  pound  of  it  worth  ?  Ans.  6s.  10|d. 

How  much  gold  of  15,  of  17,  and  of  22  caratsf  fine  must  be 
mixed  with  5oz.  of  18  carats  fine,  so  that  the  composition  may  be 
20  carats  fine  ?  Ans,  5oz.  of  15  carats  fine,  5  of  17,  and  25  of  22. 

t  A  carat  is  a  twenty  fourth  part,  22  carats  fine  means  ||  of  pure 
metal-  A  carat  is  alsa  divided  into  fonr  parts,  called  grams  of  a  carat; 


Miscellaneous  Questions.  117' 

Miscellanemis  (Questions  for  practice. 

What  number,  added  to  the  thirty-first  part  of  3813,  will  make 
the  sum  200  ?  -^ns.  77. 

The  remainder  of  a  division  is  325,  the  quotient  467,  and  the 
divisor  is  43  more  than  the  sum  of  both  ',  what  is  the  dividend  ? 

Ans.  390270. 

Two  persons  depart  from  the  same  place  at  the  same  time ; 
ihe  one  travels  SO,  the  other  35  miles  a  day  ;  how  far  are  they 
distant  at  the  end  of  7  days,  if  they  travel  both  the  same  road  ; 
and  how  far,  if  they  travel  in  contrary  directions  ? 

Jns.  35,  and  455  miles. 

A  tradesman  increased  his  estate  annually  by  lOOl.  more  than 
I  part  of  it,  and  at  the  end  of  4  years  found,  that  his  estate 
amounted  to  103421.  Ss.  9d.     What  had  he  at  first  ? 

Am.  40001. 

Divide  1200  acres  of  land  among  A,  B,  and  C,  so  that  B  may 
have  100  more  than  A,  and  C  64  more  than  B. 

Am.  A  312,  B  412,  and  C  476. 

Divide  1000  crowns  ;  give  A  120  more,  and  B  95  less,  than  C. 

Jm.  A  445,  B  230,  C  325. 

What  sum  of  money  will  amount  to  1321.  16s.  3d.  in  15 
months,  at  5  per  cent,  per  annum,  simple  interest  ?  Am.  1251. 

A  father  divided  his  fortune  among  his  sons,  giving  A  4  as 
often  as  B  3,  and  C  5  as  often  as  B  6 ;  what  was  the  whole 
legacy,  supposing  A's  share  50001.  ?  Am.  118751. 

If  1000  men,  besieged  in  a  town  with  provisions  for  5  weeks, 
each  man  being  allowed  16oz.  a  day,  were  reinforced  with  500 
men  more.  On  hearing,  that  they  cannot  be  relieved  till  the 
end  of  8  weeks,  how  many  ounces  a  day  must  each  man  have, 
that  the  provision  may  last  that  time  ?  Am.  6|oz. 

What  number  is  that,  to  which  if  f  of  |  be  added,  the  sum 
will  be  1  ?  Am.  f-f. 

A  father  dying  left  his  son  a  fortune,  |  of  which  he  spent  in  8 
months ;  ^  of  the  remainder  lasted  him  twelve  months  longer ; 
after  which  he  had  only  4101.  left.  What  did  his  father  bequeath 
him  ?  Am.  9561.  13s.  4d. 


118  »Srithmetic. 

A  guardian  paid  his  ward  35001,  for  25001.  wliicli  he  had  in 
his  hands  8  year.     >\  hat  rate  of  interest  did  he  allow  him  ? 

Jlns,  5  per  cent. 

A  person,  being  asked  the  hour  of  the  day,  said,  the  time  past 
rioon  is  equal  to  ^  of  the  time  till  midnight.    ^Yhat  was  the  time  ? 

*ins.  20min.  past  5. 

A  person,  looking  on  his  watch,  was  asked,  what  was  the  time 
of  the  day ;  he  answered,  it  is  between  4  and  5 ;  but  a  more 
particular  answer  being  required,  he  said,  that  the  hour  and 
minute  hands  were  then  exactly  together.     What  was  the  time  ? 

Jlus.  21^\  minutes  past  4. 

With  12  gallons  of  Canary,  at  6s.  4d.  a  gallon,  I  mixed  18 
gallons  of  white  wine,  at  4s.  lOd.  a  gallon  and  12  gallons  of 
cider,  at  6s.  Id.  a  gallon.  At  what  rate  must  I  sell  a  quart  of 
this  composition,  so  as  to  clear  10  per  cent.  ?       Jins.  Is.  S^-d. 

Wiiat  length  most  be  cut  off'  a  board,  8|  inches  broad,  to  con- 
tain a  square  foot ;  or  as  much  as  12  inches  in  length  and  12  in 
breadth  ?  Ms.  1 7  j|in. 

AVhat  difference  is  tliere  between  the  interest  of  3501.  at  4  pep 
cent,  for  8  years,  and  the  discount  of  the  same  sum,  at  the  same 
rate,  and  for  the  same  time  ?  Ans.  271.  S-^^^* 

A  father  devised  -/^  of  his  estate  to  one  of  his  sons,  and  ^-^  of 
the  residue  to  anotlier,  and  the  surplus  to  his  relict  for  life  ;  the 
children's  l<\giicies  were  found  to  be  2571.  3s.  4d.  different.  What 
money  did  he  leave  for  the  widow  ?  Ms.  6351.  10||d. 

.    What  number  is  that,  fi'om  whicii  if  you  take  4  of  ^,  and  to  the 
remainder  add  -^g  of  ^^g-,  tlie  sum  will  be  10  ?       Jins.  lGj\^,^^. 

A  man  dying  left  his  wife  in  expectation,  that  a  child  would 
be  afterward  added  to  the  surviving  family ;  and  making  his 
will  ordered,  that  if  the  child  were  a  son,  |  of  his  estate  should 
belong  to  him,  and  the  remainder  to  his  mother  j  but  if  it  were 
a  daugliter,  he  appointed  the  mother  |,  and  the  child  the  romain- 
dei*.  But  it  happened,  that  the  addition  was  both  a  son  and 
a  daughter,  by  which  tlie  mother  lost  in  equity  24001.  more  than 
if  it  iiad  been  only  a  daughter.  What  would  have  been  her 
ilowry,  liad  she  had  only  a  son  ?  .flm.  21001. 


JilisceUaneous  Questions,  119 

A  young  hare  starts  40  yards  before  a  grey-hound,  and  is  not 
perceived  by  him  till  she  has  been  up  40  seconds ;  she  scuds 
away  at  the  rate  of  ten  miles  an  hour,  and  tlic  dog,  on  view, 
makes  after  her  at  the  rate  of  18.  How  long  ^viIl  the  course 
continue,  and  what  will  be  the  length  of  it  from  tlie  place,  where 
the  dog  set  out  ?  Ans.  60^*^  seconds,  and  550  yards  run. 

A  reservoir  for  water  has  two  cocks  to  supply  it ;  by  the  first 
alone  it  nsay  be  filled  in  40  minutes,  by  the  second  in  50  minutes, 
and  it  has  a  discharging  cock,  by  which  it  may,  when  full,  be 
emptied  in  25  minutes.  Now  these  three  cocks  being  all  left 
open,  the  influx  and  elllux  of  the  water  being  always  at  the  same 
rate,  in  what  time  would  the  cistern  be  filled  ? 

Jlns.  3  hours  20  minutes. 

A  sets  out  from  London  for  Lincoln  precisely  at  the  time, 
when  B  at  Lincoln  sets  out  for  London,  distant  100  miles  ;  after  7 
hours  they  met  on  the  road,  and  it  then  appeared,  that  A  had 
ridden  1 A  mile  an  hour  more  than  15.  At  what  rate  an  hour  did 
each  of  them  travel  ?  Jns.  7||,  B  6||  miles. 

"What  part  of  3  pence  is  a  third  part  of  2  pence.         Jns.  |-. 

A  has  by  him  l|cwt.  of  tea,  the  piime  cost  of  which  was  961. 
sterling.  Now  interest  being  at  5  per  cent,  it  is  required  to  find 
how  be  must  rate  it  per  pound  to  B,  so  that  by  taking  his  nego- 
tiable note,  payable  at  3  months,  he  may  clear  20  guineas  by  the 
bargain?  Jns.  14s.  l]|-d.  sterlmg. 

There  is  an  island  75  miles  in  circumference,  and  3  footmen 
all  start  together  to  travel  the  same  way  about  it ;  A  goes  5 
miles  a  day,  B  8,  and  C  10 ;  when  will  they  all  come  together 
again  ?  Jns.  75  days. 

A  man,  being  asked  how  many  sheep  he  had  in  his  drove,  said, 
if  he  had  as  many  more,  half  as  many  more,  and  7  sheep  and  a 
half,  he  should  have  20  ;  how  many  had  he  ?  Jlns.  5. 

A  person  left  40s.  to  4  poor  widows.  A,  B,  C,  and  D  ;  to  A 
he  left  |,  to  B  A,  to  C  |,  and  to  D  I,  desiring  the  whole  might 
be  distributed  accordingly ;  what  is  the  proper  share  of  each  ? 

Ms.  A's  share  14s.  ^fd.  B's  10s.  6^|d.  C's  8s.  5/^d.  D's 
7s.  ,V1- 

A  general,  disposing  of  his  army  intq  a  squaie.  finds  he  has 


ISO  Arithmetics 

284  soldiers  over  and  above  ;  but  increasing  each  side  with  one 
soldier,  he  wants  25  to  fill  up  the  square ;  how  many  soldiers 
had  he  ?  wins.  24000. 

There  is  a  prize  of  2121.  14s.  7d.  to  be  divided  among  a  cap- 
tain, 4  men,  and  a  boy ;  the  captain  is  to  have  a  share  and  a 
half;  the  men  each  a  share,  and  the  boy  |  of  a  share;  what 
ought  each  person  to  have  ? 

Ans.  Tiie  captain  541.  14s.  ^d.  each  man  361.  9s.  4|d.  and  the 
boy  121.  3s.  13(1. 

A  cistern,  containing  60  gallons  of  water,  has  3  unequal  cocks 
for  discharging  it;  the  greatest  cock  will  empty  it  in  one  hour, 
the  second  in  2  hours,  and  the  third  in  3  ;  in  what  time  will  it  be 
emptied,  if  they  all  run  together  ?  Ans.  S2^j  minutes. 

In  an  orchard  of  fruit  trees,  i  of  them  bear  apples,  |  pears,  | 
plums,  and  50  of  them  cherries  :  how  many  trees  are  there  in 
all  ?  Jns.  600. 

A  can  do  a  piece  of  work  alone  in  ten  days,  and  B  in  13;  if 
both  be  set  about  it  together,  in  what  time  will  it  be  finished  ? 

dns.  5l|  days. 

A,  B,  and  C  are  to  share  lOOOOOl.  in  the  proportion  of  4.  |, 
and  \.  respectively ;  but  C's  part  being  lost  by  his  death,  it  is 
required  to  divide  the  whole  sum  properly  between  the  other  two. 
Am.  A's  part  is  57142f  1.  and  B's  4285711. 


APPENDIX, 

CONTAINING  TABLES  OF  VARIOUS    WEIGHTS    AND  MEASURES. 


Measures. 

The  weights  and  measures  in  common  use  arc  liable  to  great 
uncertainty  and  inconvenience.  There  being  no  fixed  standard  at 
hand;  by  which  their  accuracy  can  be  ascertained,  a  great  variety 
of  measures,  bearing  the  same  name,  has  obtained  in  different 
countries.  The  foot,  for  instance,  is  used  to  stand  for  about  a 
hundred  different  established  lengths.  The  several  denomina- 
tions of  weights  and  measures,  are  also  arbitrary,  and  occasion 
most  of  the  trouble  and  perplexity,  that  learners  meet  with  in 
mercantile  arithmetic. 

To  remedy  these  evils,  the  French  government  adopted  a 
new  system  of  weights  and  measures,  the  several  denomina- 
tions of  which  proceed  in  a  decimal  ratio,  and  all  referable  to  a 
common  permanent  standard,  established  by  nature,  and  acces- 
sible at  all  places  on  the  earth.  This  standard  is  a  meridian  of 
the  earth,  which  it  was  thought  expedient  to  divide  into  40  mil- 
lion parts.  One  of  these  parts  is  assumed  as  the  unit  of  length, 
and  the  basis  of  the  whole  system.  This  they  called  a  metre. 
It  is  equal  to  about  39|English  inches,  of  which  submultiples 
and  multiples  being  taken,  the  various  denominations  of  length 
are  formed. 

Kng.  Inch  Dec. 
,0393/ 

,39371 

3,93710 

39,37100 

393,71000 

3937,10000 

39371,00000 

393710,00000 


A  millimetre  is  the  1000th  part  of  a  metre 
A  centimetre     tlie  100th  part  of  a  metre 
A  decimetre       the  10th  part  of  a  metre 
A  METRE 
A  decametre 
A  hecatometre 
A  chiliometre 
A  myriometre 


10        metres 

100       metres 

1000     metres 

10000  metres 
A  grade  or  degree  of  the  meridian  equaj  to 
100000  metres,  or  lOOtli  part  of  the  quadrant. 

16 


3937100,00000 


Mis. 

Fur. 

Yds. 

Ft. 

In.De. 

0 

0 

10 

2 

9,7 

0 

0 

109 

1 

1 

0 

4 

213 

1 

10,2 

6 

1 

156 

0 

6 

132  Jlppendix, 

The  decametre        is 

The  iiecatometre 

The  chiliometre 

The  myriomctre 

The  grade  or  decimal  degree  of  the 

meridian  62     1       23     2       8 

Measures  of  Capacity. 
A  cube,  whose  side  is  one  tenth  of  a  metre,  that  is,  a  cubic 
decimetre,  constitutes  the  unit  of  measures  of  capacity.     It  is 
called  the  litre,  and  contains  61,028  cubic  inches. 

Eng.  Cub.  In.  Sec. 

A  niillilitre  or  1000th  part  of  a  litre  ,06103 

A  centilitre       100th  of  a  litre  ,61028 

A  decilitre         10th  of  a  litre  6,10280 

A  litre,  a  cubic  decimetre'  61,02800 

A  decalitre        10  litres  610,28000 

A  hecatolitre     1000  litres  6102,80000 

A  chiliolitre      10000  litres  61028,00000 

A  myriolitre      100000  litres  610280,00000 

The  English  pint,  wine  measure,  contains  28,875  cubic  inches. 
The  litre  therefore  is  2  pints  and  nearly  one  eighth  of  a  pint. 

Hence 
A  decalitre  is  equal  to  2  gal.  64 ^W  cubic  inches. 
A  hecatolitre  26  gal.  A^-^\  cubic  inches. 
A  chiliolitre  264  gal.  ^Yt  cubic  inches. 

Weights. 

The  unit  of  weight  is  tlie  gramme.  It  is  the  weight  of  a  quan- 
ity  of  pure  water,  equal  to  a  cubic  centimetre,  and  is  equal  to 
15,444  grains  Troy. 

Gr.  Dee. 

A  milligramme  is  1000th  part  of  a  gramme  0,0154 

A  centigramme  100th  of  a  gramme  0,1544 

A  decigramme  10th  of  a  gramme  1,5444 

A  gramme,  a  cubic  centimetre  15,444o 

A  decagramme     10         grammes  154,4402 

A  hecatogramrae  100      grammes  1544,402S 


Mw  French  Weights  and  Measures.  123 

A  chilogramme    1000    grammes  15444,0234 

A  myriogramme  10000  grammes  154440,2344 

A  gramme  being  equal  to  15,444  grains  Troy. 
A  decagramme  6dwt,  10,44gr.  equal  to  5,65  drams  Avoirdupois. 

lb.     oz.  dr. 

A  hecatogramme      equal  to  0     3       8,5  avoird. 

A  cliilogramme  2     3       5  avoird. 

A  myriogramme  22     1     15  avoird. 

100  myriogramms  make  a  tun,  wanting  S2lb.  8oz. 

Land  Measure. 

The  unit  is  the  are,  which  is  a  square  decametre,  equal  to  3,^5 
perches.  The  deciare  is  a  tenth  of  an  are,  the  centlare  is  100th 
of  an  are,  and  equal  to  a  square  metre.  The  milliare  is  1000th 
of  an  are.  The  decare  is  equal  to  10  ares  ;  the  hecatare  to  100 
ares,  and  equal  to  2  acres  1  rood  35,4  perches  English.  The 
chilare  is  equal  to  1000  ares,  the  myriare  to  10000  ares. 

For  fire-wood  tlie  stere  is  the  unit  of  measure.  It  is  equal  to 
a  cubic  metre,  containing  35,3171  cubic  feet  English.  The  de- 
cestere  is  the  tenth  of  a  stere. 

The  quadrant  of  the  circle  generally  is  divided  like  the  fourth 
part  of  the  meridian,  into  100  degrees,  each  degree  into  100 
minutes  and  each  minute  into  100  seconds,  &c.  so  that  the  same 
number,  which  expresses  a  portion  of  the  meridian,  indicates 
also  its  length,  which  is  a  great  convenience  in  navigation. 

The  coin  also  is  compreliended  in  this  system,  and  made  to 
conform  to  their  unit  of  weight.  The  weight  of  the  franco  of 
which  one  tenth  is  alloy,  is  fixed  at  5  grammes  j  its  tenth  part  is 
called  decime,  its  hundredth  part  centime. 

The  divisions  of  time,  soon  after  the  adoption  of  the  above,  un- 
derwent a  similar  alteration. 

The  year  was  made  to  consist  of  12  months  of  30  days  each, 
and  the  excess  of  5  days  in  common  and  6  in  leap  years  was  con- 
sidered as  belonging  to  no  month.  Each  .month  was  divided 
into  three  parts,  called  decades.  Tlie  day  was  divided  into  10 
hours,  each  hour  into  100  minutes,  and  each  minute  into  100 
seconds.     This  new  calendar  was  adopted  in  1 793 ;  in  1 805  it 


124 


*Spp€nd'ix, 


was  abolished,  aud  the  old  calender  restored.  The  weights  and 
measures  are  still  in  use,  and  will  probably  pi-evail  through- 
ought  the  continent  of  Europe.  They  are  recommended  to  the 
attention  of  every  civilized  country  ;  and  their  general  adoption 
would  prove  of  no  small  importance  to  the  scientific,  as  well  as 
the  commercial  world. 


Scripture  Long  Measure* 


4t 

Digit 

... 

Feet.    In.  Dec. 
0,912 

3 

Palm 

0 

3,648 

2 

Span 

0 

10,944 

4 

Cubit 

1 

9,888 

H 

Fathom 

Y 

5,552 

H 

Ezekiel's  reed 

10 

11,328 

10 

Arabian  pole 

14 

7,104 

Scoenus,  measuring  line 

145 

1,104 

N.  B.  There  was  another  span  used  in 

the  East,  et^ual  to  ith 

of  a  cubit. 

Qrecian  Long  Measure  reduced  to  English. 

Eng.  paces.  Feet. 

In,    Dec. 

4 

Dactylis,  Digit 

0     0 

0,75  54j|- 

n 

Doron,  Dochrae,  Palesta, 

0     0 

3,021 8| 

ItV 

Lichas 

0     0 

7,5546|. 

ItV 

Orthodoron 

0     0 

8,3101-«^ 

n 

Spithame 

0     0 

9,065Ui 

H 

Pous,  foot 

0     1 

0,0875 

n 

Pygme,  cubit 

0     1 

1,5984| 

n 

Pygon 

0     1 

3,1 09| 

4 

Pecus,  cubit  larger 

0      1 

6,13125 

100 

Orgya,  pace 

0     6 

0,525 

8 

Sta^™^  furlong 

100     4 

4,5 

Million,  Mile 

805      5 

0 

N.  B.  Two  sorts  of  long  measures  were  used  in  Greece,  viz. 
the  Olympic  and  the  Pythic.  The  former  was  used  in  Pelopon- 
nesus, Attica,  Sicily,  and  the  Greek  cities  in  Italy.  The  latter 
was  used  in  Thessaly,  lUyria,  Phocis,  and  Thrace. 

t  These  numbers  show  how  many  of  each  denomination  it  takes  to 
make  one  of  the  next  Ibilowing. 


Tables  of  7 f eights  and  Measures, 


125 


The  Olympic  foot,  properly  called  the  Greek  foot,  according  t» 
Dr.  Hiitton,  contains     12,108  English  inches, 
Folkes,  12,072 

Cavallo,  12,084 

Tlie  Pythic  foot,  called  also  natural  foot,  according  to 
Hutton,  contains       9,768 
Paucton,  9,731 

Hence  it  appears,  that  the  Olympic  stadium  is  201|  English 
yards  nearly  ;  and  the  Pythic  or  Delphic  stadium,  162}  yards 
nearly  ;  and  the  other  measures  in  proportion. 

The  Phyleterian  foot  is  the  Pythic  cubit,  or  1|  Pythic  foot. 
The  Macedonian  foot  was  13,92  English  inches  ;  and  the  Siciliaa 
foot  of  Archimedes,  8,76  English  inches. 


Jewish  Long  or  Itinerary  Measure, 


Eng,  Miles.     Paces.  Feet.  Dec*. 

400 

Cubit 

0          0      1,824 

5 

Stadium 

0     145     4,6 

2 

Sabbath  day's  journey 

0     729  "  3,0 

3 

Eastern  mile 

1     403      1,0 

8 

Parasang 

4      153     3,0 

A  day's  journey 

33     172     4,0 

Roman  Long  Measures  reduced 

to  English. 

Bug:.  Paces.   Feet.  In.  Dec 

H 

Digitus  traversus 

0       0       0,725| 

3 

Uncia,  or  Inch 

0     0     0,967 

4 

Palma  minor 

0     0     2,901 

11 

Pes,  or  Foot 

0     0   11,604 

H 

Pal  mi  pes 

0      1      2,505 

H 

Cubitus 

0     1     5,406 

2 

Gradus 

0     2     5,01 

125 

Passus 

0     4   10,02 

8 

Stadium 

120     4     4,5 

Milliare 

967     0        0 

N.  B.  The  Roman  measures  began 

with  6  scrupula  =  1  sicili- 

wim  ;  8  scrupula  =  1  duellum  ;  1|  diicUu 

n  =  1  seminaria ;  and 

1 8  scrupuj 

\  =  1  digitus.  Two  passus  were  equal  to  1  decenipeda. 

Ha 


Appendix. 


Attic  Dry  Measures  reduced  to  English, 


Pecks.  Gall. 

Pints. 

Sol.  Inch. 

10 

Cochliarion 

0      0 

0 

0,27  6/y 

H 

Cyathus 

0      0 

0 

2,763| 

4 

Oxybaphon 

0      0 

0 

4,144| 

2 

Cotylus 

.     0      0 

0 

16,579 

H 

Xestes,  sextary 

0     0 

0 

33,158 

48 

Choenix 

0     0 

1 

15,7051 

Medimnus 

4     0 

6 

3,501 

Attic  Measures  of  Capacity  for  Liquids,  reduced  to  English  Wine 
Measure. 


Gal. 

Pints.  Sol.  In.    Dec. 

2 

Cochliarion 

0 

^1,     0,0356/^ 

H 

Cheme 

0 

io        O'OnSf 

2 

Myston 

0 

ji^       0,089^1 

2 

Concha 

0 

A       0,1781-i 

H 

Cyathus 

0 

^\       0,35611 

4 

Oxybathon 

0 

i        0,5353 

2 

Cotylus 

0 

i        2'1411 

6 

Xestes,  sextary 

0 

1        4,283 

12 

Chous,  congius 

0 

6     25,698 

Metretes,  amphora 

10 

2     19,-626 

Others 

reckon  6  choi  =  1  amphoreus, 

and  2 

amphorei  =  1 

keramion 

or  metretes.     The  keramion  is 

stated 

by  Paucton  to 

have  been 

equal  to  35  French  pints, 

or8| 

El 

iglish  gallons,  and 

the  other 

measures  in  proportion. 

Measures 

of  Capacity  for  Liquids, 

reduced 

to 

English    Wine 

Measure. 

Gal. 

Pints.  Sol.  In.  Dec. 

4 

Ligula 

0 

:rV     0,1.7/^ 

H 

Cyathus 

0 

tV     0,469| 

2 

Acetabulum 

0 

i      0,7041 

2 

Quartarius 

0 

i      1,409 

2 

Hemina 

0 

1      2,818 

6 

Sextarius 

,0 

1       5,636 

4 

Congius 

0 

7      4,942 

2 

Urna 

3 

41     5,33 

90 

Amphora 

7 

1     10,66 

Culeus 

143 

3     11,095 

\ 


Tables  of  Weights  and  Measures. 


127 


Jewish  Dry  Measures  reduced  to  English. 


20 

Gachal 

1^ 

Cab 

H 

Gomor 

3 

Seah 

5 

Epha 

2 

Letteeh 

C  homer,  coron 

Pecks.  Gal. 
0       0 


0 
0 
1 

3 
16 

32 


nt  1"* 


Sol.  Inch, 
0,031 

0,073 

1,211. 

4,036 

12,107 

26,500 

18,969 


Jewish  Measures  of  Capacity  far  Liquids,  reduced  to  English  Wine 
Measure. 


4 
3 
2 
3 
10 


Capli 

Log 

Cab 

Hin 

Seah 

Bath,  epha 

Coron,  chomer 

Ancient  Roman  Land  Measure. 


Gal.  Pints.  SoL  Inch. 
0 
0 
0 
1 

2 
7 

75 


0,177 
0,211 
0,844 
2,533 
5,067 
15,2 
7,625 


100  Square  Roman  feet 

4  Scrupula 
1^  Sextulus 

6  SextuU  or  5  Actus 

6  Uticise 

2  Square  Actus 

2  Jugera 
100  Heredia 


=  1  Scrupulum  of  land 

=  1  Sextulus 

=  1  Actus 

=  1  Uncia  of  land 

=  1  S(iuare  Actus 

=  1  Jugerum 

=  1  Heredium 

=  1  Centuria 


N.  B.  If  we  take  the  Roman  foot  at  11,6  English  inches,  the 
Roman  jugerum  was  5980  English  square  yards,  or  1  acre  37| 
perches. 

Roman  Dry  Measures  reduced  to  English. 


Peck.  Gal.  Pint.  Sol.  In.  De. 


Ligula 

Cyathus 

Acetabulum 

Hemina  or  Trutta 

Sextarius 

Semi  d. 

Modius 


1 

0 
0 


0,01 
0,04 
0,06 
0,24 
0,48 
3,84 
7,68 


42" 


a 

i 

s^ 

p^  s 

= 

> 

il 

sv. 

c 

fc^  §^ 

.."i; 

;5a- 

1 

;r^ 

u 

ft^ 

II 


1^ 


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w  "o 

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£^, 

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GENERAL  LIBRARY -U.C.  BERKELEY 


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